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Maximal regularity in weighted spaces, nonlinear boundary conditions, and global attractors [Elektronische Ressource] / von Martin Meyries

221 pages
Maximal Regularity in Weighted Spaces,Nonlinear Boundary Conditions,and Global AttractorsZur Erlangung des akademischen Grades einesDOKTORS DER NATURWISSENSCHAFTENvon der Fakultät für Mathematik desKarlsruher Instituts für TechnologiegenehmigteDISSERTATIONvonMartin Meyriesaus Germersheim am RheinTag der mündlichen Prüfung: 24. November 2010Referent: Prof. Dr. Roland SchnaubeltKorreferenten: Prof. Dr. Jan Prüss, Prof. Dr. Lutz WeisFür meine ElternContentsIntroduction 11 The Spaces L and Weighted Anisotropic Spaces 11p; 1.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Abstract Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.2.1 Abstract Maximal L -Regularity . . . . . . . . . . . . . . . . . . . 28p; 1.2.2 Operator-Valued Fourier Multipliers . . . . . . . . . . . . . . . . . . 311.3 Weighted Anisotropic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 321.3.1 The Newton Polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.3.2 Temporal Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.3.3 Spatial Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.3.4 Pointwise Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 Maximal L -Regularity for Static Boundary Conditions 65p; 2.1 The Problem and the Approach in Weighted Spaces . . . . . . . . . . . . . 65n n2.
Voir plus Voir moins

MaximalRegularityinWeightedSpaces,
Conditions,BoundaryNonlinearttractorsAGlobaland

ZurErlangungdesakademischenGradeseines

WISSENSCHAFTENTURNADERDOKTORS

vonderFakultätfürMathematikdes
KarlsruherInstitutsfürTechnologie
genehmigte

TIONATDISSER

onv

MartinMeyries

RheinamGermersheimaus

TagdermündlichenPrüfung:24.November2010

Referent:Prof.Dr.RolandSchnaubelt

Korreferenten:Prof.Dr.JanPrüss,Prof.Dr.LutzWeis

Für

meine

Eltern

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1

1TheSpacesLandWeightedAnisotropicSpaces11
p,µ1.1BasicProperties.................................11
1.2AbstractProperties................................28
1.2.1AbstractMaximalL-Regularity...................28
p,µ1.2.2Operator-ValuedFourierMultipliers..................31
1.3WeightedAnisotropicSpaces..........................32
1.3.1TheNewtonPolygon...........................34
1.3.2TemporalTraces.............................38
1.3.3SpatialTraces...............................44
1.3.4PointwiseMultipliers...........................50

2MaximalLp,µ-RegularityforStaticBoundaryConditions65
2.1TheProblemandtheApproachinWeightedSpaces.............65
nn2.2TopOrderConstantCoefficientOperatorsonRandR..........72
+2.2.1TheFull-SpaceCasewithoutBoundaryConditions..........72
2.2.2TheHalf-SpaceCasewithBoundaryConditions............73
2.3TopOrderCoefficientshavingSmallOscillation................79
2.4TheGeneralCaseonaDomain.........................86
2.5ARight-InversefortheBoundaryOperator..................98

3MaximalL-RegularityforBoundaryConditionsofRelaxationType113
p,µ3.1TheProblemandtheApproachinWeightedSpaces.............113
3.2Half-SpaceProblemswithBoundaryConditions................125
3.2.1ConstantCoefficients...........................125
3.2.2TopOrderCoefficientshavingSmallOscillation............134
3.3TheGeneralCaseonaDomain.........................138

4AttractorsinStrongerNormsforRobinBoundaryConditions143
4.1Introduction....................................143
4.2SuperpositionOperators.............................146
4.3TheLocalSemiflow................................153
4.4GlobalAttractorsinStrongerNorms......................160

4.5Applications....................................164
4.5.1Reaction-DiffusionSystemswithNonlinearBoundaryConditions..164
4.5.2AChemotaxisModelwithVolume-FillingEffect...........165
4.5.3APopulationModelwithCross-Diffusion...............166

5BoundaryConditionsofReactive-Diffusive-ConvectiveType167
5.1Introduction....................................167
5.2MaximalL-RegularityfortheLinearizedProblem.............169
p,µ5.3TheLocalSemiflowforQuasilinearProblems.................173
5.4APrioriHölderBoundsimplyGlobalExistence................175
5.5TheGlobalAttractorforSemilinearDissipativeSystems...........180

191endixAppAnA.1BoundariesofDomainsinR..........................191
A.2InterpolationTheory...............................192
A.3SectorialOperators................................194
A.4FunctionSpacesonDomainsandBoundaries.................201
A.5DifferentialOperatorsonaBoundary......................205
A.6Gagliardo-NirenbergInequalities........................207

yBibliograph

209

ductiontroIn

Thesubjectofthisthesisisthemathematicalanalysisoflinearandquasilinearparabolic
problemswithinhomogeneousandnonlinearboundaryconditions.Weconsiderstatic
boundaryconditionsofDirichlet,NeumannorRobintype,andfurtherboundarycon-
ditionsofrelaxationtype,whichincludedynamiconesaswellasboundaryconditionsthat
ariseinthelinearizationoffreeboundaryproblems.

Evolutionequationsofthistypedescribeagreatvarietyofphysical,chemicalandbiological
phenomena,likereaction-diffusionprocesses,phasefieldmodels,chemotacticbehaviour,
populationdynamics,phasetransitionsandthebehaviouroftwophasefluids,forinstance.
Inmanycasesitisnecessarytoimposenonlinearboundaryconditionsintoareaction-
diffusionmodeltocapturethedynamicsofthephenomenonunderinvestigation.Inthe
contextoffreeboundaryproblemsnonlinearboundaryconditionsnaturallyariseaftera
transformationtoafixeddomain.

WefocusonmaximalregularityresultsinweightedLp-spacesforlinearnonautonomous
parabolicproblemswithinhomogeneousboundaryconditions.Comparedtotheapproach
withoutweights,weareabletoreducethenecessaryregularityoftheinitialvalues,to
incorporateaninherentsmoothingeffectintothesolutionsandtoavoidcompatibility
conditionsattheboundary.Thesepropertiesserveusasabasisforconstructingalocal
semiflowforthecorrespondingquasilinearproblemsinascaleofphasespaces,andforthe
investigationofthelong-timebehaviourofsolutionsintermsofglobalattractors.

Ourapproachtoquasilinearproblemsthusreliesonlinearizationandagoodunderstanding
ofthelinearproblem.ThisideagoesbackatleasttoKato[58],Sobolevskii[77]and
Solonnikov[79].InasemigroupcontextitwascarriedoutbyGrisvard[46],DaPrato&
Grisvard[22],Amann[3,4,5,6,7],DaPrato&Lunardi[23],Lunardi[67]andPrüss[70].
Semilinearproblemscanbetreatedintheframeworkofanalyticsemigroups,seeHenry’s
[51].monograph

Maximalregularitymeansthatthereisanisomorphismbetweenthedataandthesolution
ofthelinearprobleminsuitablefunctionspaces.Havingestablishedsuchasharpregularity
resultforthelinearization,thecorrespondingquasilinearproblemcanbetreatedbyquite
simpletools,likethecontractionprincipleandtheimplicitfunctiontheorem.Thereare
approachesinspacesofcontinuousfunctions(seeAngenent[12]andClément&Simonett
[19]),inHölderspaces(seeLunardi[67])andinLp-spacesforp∈(1,∞)(seeClément

2

ionuctdIntro

&Li[17]andPrüss[70]).Formoredetailsandotherapproachestoquasilinearparabolic
problemswerefertothediscussionin[10].
Thethreementionedmaximalregularitysettingshaveadvantagesanddisadvantages.The
continuoussettingisquitesimple,butstrongrestrictionsontheunderlyingspacesare
necessary.IntheHöldersettingthenonlinearitiesareeasytohandleandtheapproach
isalsoapplicabletofullynonlinearproblems,butunpleasantcompatibilityconditionsat
theinitialtimearenecessaryandaprioriestimatesinhighnormsarerequiredtoshow
globalexistenceofsolutions.IntheLp-settingpowerfultoolsfromvector-valuedharmonic
analysisareavailable(andneeded!),butontheotherhandgeometricassumptionsonthe
underlyingspacesarerequiredandalsohereonehastoworkinhighnorms.Forafurther
discussionwereferagainto[10].InthisthesisweentirelyworkinanLp-framework.
TodecidewetheraconcretelinearproblemenjoysmaximalLp-regularityinasuitable
settingisnoteasy.Forlinearproblemswhichcanbereducedtoanabstractequationof
formthe∂tu(t)+Au(t)=f(t),t>0,u(0)=u0,(1)
onaBanachspaceE,whereAisaclosedanddenselydefinedoperatoronE,theoperator
summethod,asdevelopedbyDaPrato&Grisvard[21]andextendedbyDore&Venni[31]
andKalton&Weis[57],isappropriateinmanycases.Weis[85]characterizedthemaximal
Lp-regularitypropertiesofanoperatorintermsofR-sectoriality.IfEisaHilbertspace,
theneverynegativegeneratorofaboundedanalyticC0-semigroupenjoysmaximalLp-
regularity.Unfortunately,aHilbertspacesettingdoesoftennotseemtobesuitableforthe
applicationstoquasilinearproblems.
Totreatsecondorderparabolicdifferentialequationswithinhomogeneousornonlinear
boundaryconditionsinamaximalLp-regularityapproachonetypicallychoosesE=Lp,
E=Wp−1orEasaninterpolationspaceinbetweenasabasicunderlyingspace.IfEis
closetoWp−1thentheboundaryconditionsareapriorionlysatisfiedinaweaksense,butin
thiswaytheproblemcanbecastintheform(1)andoperatorsummethodsareavailable,
inprinciple.IfEisclosetoLp,thentheboundaryconditionscanbeunderstoodina
pointwisesense,butaformulationintheabstractform(1)doesnotseemtobepossiblein
areasonableway,ingeneral-thereisalwaysa’PDEpart’lefttodealwith.Anadvantage
ofchoosingEclosetoLpisthatgrowthconditionsonthenonlinearitiescanbeavoided.
Combiningoperatorsummethodswithtoolsfromvector-valuedharmonicanalysis,Denk,
Hieber&Prüss[24,25]andDenk,Prüss&Zacher[26]showedmaximalLp-regularitywith
Lpasanunderlyingspaceforalargeclassofvector-valuedparabolicproblemsofevenorder
withinhomogeneousboundaryconditions.In[25]problemswithboundaryconditionsof
statictypeareconsidered,i.e.,
∂tu+A(t,x,D)u=f(t,x),x∈Ω,t>0,
Bj(t,x,D)u=gj(t,x),x∈Γ,t>0,j=1,...,m,(2)
u(0,x)=u0(x),x∈Ω.

tioncduIntro

3

Thisincludesthelinearizationofreaction-diffusionsystemsandofphasefieldmodelswith
Dirichlet,NeumannandRobinconditions.In[26]theauthorsstudyproblemswithbound-
aryconditionsofrelaxationtype,i.e.,
∂tu+A(t,x,D)u=f(t,x),x∈Ω,t>0,
∂tρ+B0(t,x,D)u+C0(t,x,DΓ)ρ=g0(t,x),x∈Γ,t>0,
Bj(t,x,D)u+Cj(t,x,DΓ)ρ=gj(t,x),x∈Γ,t>0,j=1,...,m,(3)
u(0,x)=u0(x),x∈Ω,
ρ(0,x)=ρ0(x),x∈Γ,
whichincludesdynamicboundaryconditionsaswellasproblemsarisingaslinearizations
offreeboundaryproblemsthataretransformedtoafixeddomain.HereΩ⊂Rnisa
domainwithcompactsmoothboundaryΓ=∂Ω.Thecoefficientsoftheoperatorsareonly
assumedtobepointwisemultiplierstotheunderlyingspaces,andthetopordercoefficients
arerequiredtobeboundedanduniformlycontinuous.Theseregularityassumptionsallow
toapplythelinearresultstoquasilinearproblems.Earlierinvestigationson(2)startedat
leastwithLadyzhenskaya,Solonnikov&Ural’ceva[64]andincludealsoWeidemaier[84].
AprincipleshortcomingofthemaximalLp-regularityapproachto(1),(2)and(3)isthat
forfixedponecansolvetheequationforinitialvaluesonlyinonesinglespaceofrelatively
highregularity,andthatonedoesnothavetheflexibilitytoworkinascaleofspaces.
TheLp-approachto(1)necessarilyrequiresthatu0belongstotherealinterpolationspace
(E,D(A))1−1/p,p.Forlargep,whichisnecessarytochooseintheLp-settingtoensurethat
thenonlinearitiesarewell-defined,thisspaceisclosetothedomainofA.Thesituationfor
(2)and(3)issimilar.Thusthelong-timebehaviourofsolutionsmustbeinvestigatedina
phasespaceofhighregularity.
Forsecondorderproblems(E,D(A))1−1/p,pisusuallyclosetoWp2forlargep,butoften
thestructureoftheproblemsunderconsiderationdoesnotprovideenoughinformation
foraprioriestimatesinsuchhighnorms.Suchestimatesaretypicallyobtainedinthe
energyspaceH21,inL∞orinaHölderspaceCαwithsmallexponent.Thusthereisagap
betweentheregularitiesinherenttogivenproblemsandtheregularitieswhicharenecessary
toapplythenonlineartheorybasedonmaximalLp-regularity.Duetothelackofascaleof
phasespacesitisfurthernotclearhowtoshowrelativecompactnessofboundedorbitsand
compactnessofthesolutionsemiflowwithoutstrongaprioribounds.Thelatterproperties
areimportantintheinvestigationoftheω-limitsetofsolutionsandinthecontextofglobal
attractors.ThesituationisevenworseforthemaximalHölderregularityapproach.Hereitisre-
quiredthattheinitialvaluesbelongtothedomainoftheoperatorunderconsideration.
Ontheotherhand,forsemilinearproblemsthedomainsoffractionalpowersofopera-
torsserveasanaturalscaleofphasespaces.Theapproachtoquasilinearproblemsin
interpolation-extrapolationscalesdevelopedbyAmannalsodoesnothavetheseshortcom-
ings,butrequiresthattheboundaryconditionscanbeabsorbedintothedomainofan
operatoronanegativeorderbasespace.

4

ionuctdIntro

ToclosethisregularitygapbetweentheoryandapplicationsinthemaximalLp-regularity
approachonehasintroducedtemporalweightsthatvanishattheinitialtime.Inanabstract
settingthiswasdonebyClément&Simonett[19]inthecontextofcontinuousmaximal
regularity,andbyPrüss&Simonett[71]intheLp-setting.Thelatterauthorsproposedto
workinthepowerweightedspaces
Lp,µ(R+;E)=u:R+→E:tp(1−µ)|u(t)|Epdt<∞,
R+whereµ∈(1/p,1].(Notethattheweightstp(1−µ)belongtotheclassAp,seeStein[81].)
Functionswithworsebehaviouratt=0belongtoLp,µifonelowersµ.Thisapproachyields
thesolvabilityoftheabstractequation(1)forinitialvaluesin(E,D(A))µ−1/p,p,andthus
allowstoreducetheinitialregularityuptotheunderlyingspaceE.Forfixedpthisfurther
givesausefulscaleofspacesfortheinitialvalues.Sincetheweightstp(1−µ)onlyhavean
effectatt=0(onfinitetimeintervals),themaximalregularityapproachintheLp,µ-spaces
alsoprovidesaninherentsmoothingeffectintosolutions,astheyregularizeinstantaneously
from(E0,E1)µ−1/p,pto(E0,E1)1−1/p,p,whichcorrespondstotheunweightedcaseµ=1.
Itwasfurthershownin[71]thatthepropertyofmaximalLp,µ-regularityforaclosedand
denselydefinedoperatorisindependentofµ∈(1/p,1].Hencetheoperatorsummethods
knownfromtheunweightedcasearealsoavailableintheweightedapproach.Theresults
of[71]wererecentlyusedbyKöhne,Prüss&Wilke[59]toestablishadynamictheoryfor
problems.quasilinearabstractItisthemainpurposeofthepresentthesistoextendandcombinetheresultsof
[25,26,59,71]describedaboveandtodevelopthemaximalLp,µ-regularityapproachfor
theproblemclasses(2)and(3).Hereweaimatasystematicandcomprehensivetreatment
ofthesolutiontheoryaswellasofthevariousprerequisitessuchastraceandinterpolation
propertiesoftheunderlyinganisotropicfunctionspacesonspace-time.Besidesthereduc-
tionoftheinitialregularityandaninherentsmoothingeffectofsolutions,theapproach
allowstoavoidcompatibilityconditionsattheboundaryinlinearproblems.Weapply
ourlineartheorytoquasilinearreaction-diffusionsystemswithnonlinearboundarycondi-
tions,ofRobinandofreactive-diffusive-convectivetype,respectively.Forsuchproblems
weinvestigatelocalwell-posednessinascaleofphasespaces,globalexistenceandglobal
attractors,employingtheflexibilityofmaximalLp,µ-regularity.
Wedescribetheorganizationofthethesis,themainresultsandthemethodswehave
used.InChapter1weinvestigatethevector-valuedLp,µ-spacesandthecorresponding
anisotropicSobolev-Slobodetskiispacesinasystematicway,anddeducealltheproperties
requiredfortheapplicationstoparabolicproblems.Forinstance,spacesoftype
Wκp,µR+;Lp(Γ)∩Lp,µR+;Wp2κ(Γ),
whereκ∈(0,1),naturallyariseintheLp-approachto(2)and(3)asthesharpregularity
classoftheboundaryinhomogeneities.Forsuchspacesweestablishanintrinsicnorm,var-
iousembeddingsviatheNewtonpolygon,thepropertiesofthetemporalandthespatial

tioncduIntro

5

tracesandmappingpropertiesofpointwisemultipliers.Sincethemultiplicationwiththe
weightisnotanisomorphismtotheunweightedSobolev-Slobodetskiispacesmostofthe
propertiescannotbededucedfromknownresults.Wemainlyemployinterpolationtech-
niques,operatorsummethodsandtherepresentationofthespacesasdomainsofoperators
withaboundedH∞-calculusorboundedimaginarypowers.Ourexpositionalsogivesa
comprehensiveaccountoftheunweightedcase(µ=1),whichhasbeentreatedintheliter-
aturesofaronlyinascatteredway.Itturnsoutthattheweightedspacesenjoyanalogous
propertiesastheunweightedspaces,exceptfortheintendedreducedregularityoftracesat
t=0,ofcourse.Thismakestheweightedsettingapplicabletoparabolicproblemswithout
tages.andisadvCertainaspectsofweightedfractionalorderspaceswerealreadyinvestigatedbyGrisvard
[44],Triebel[82]andPrüss&Simonett[71].RecentlyGirardi&Weis[42]showedan
operator-valuedFouriermultipliertheoremfortheLp,µ-spaces.

Buildingonthepropertiesoftheweightedspaces,inChapters2and3wegeneralizetothe
Lp,µ-settingthemaximalregularityresultsbyDenk,Hieber&Prüss[25]andDenk,Prüss
&Zacher[26]onvector-valuedlinearinhomogeneous,nonautonomousinitial-boundary
valueproblemsoftheform(2)and(3).TheunknownstakevaluesinaBanachspaceof
classHT,whichisnecessarytoapplytoolsfromharmonicanalysis,andweimposethe
sameellipticityandLopatinskii-Shapiroconditionsontheoperatorsasintheunweighted
case.Againthecoefficientsoftheoperatorsareonlyrequiredtobepointwisemultipliers
ontheunderlyingspaces,withcontinuoustopordercoefficients,whichallowstoapplythe
problems.quasilineartotheorylinearTheChapters2and3areorganizedanalogously.InSections2.1and3.1wegiveadetailed
descriptionoftheapproachandtheinvolvedfunctionspaces,provideexamplesandfor-
mulatethepreciseassumptions,respectively.ThemainresultsarestatedintheTheorems
2.1.4and3.1.4.Theirproofs,whichareinspiredbytheonesin[25,26],isthencarried
outintherestofthechapters.IntheSections2.2and3.2thecaseoffull-andhalf-space
constantcoefficientmodelproblemswithoutlowerordertermsareconsidered.Herewe
employtoalargeextentthepropertiesoftheweightedspacesderivedinChapter1.Since
theseresultsenterinallpointsofthereasoning,wegivethelongandtechnicalproofin
detail.Therestofthechaptersisthendevotedtoaperturbationandlocalizationproce-
duretoderivethecaseofageneraldomainfromthemodelproblems.Thisprocedureis
againquitetechnical,inparticularbecauseonehastotakecaretocontroltheconstantsin
thevariousperturbationsteps.InProposition2.5.1wealsoshowthatboundaryoperators
relatedto(2)aresurjectiveonsuitablefunctionspacesandhaveaboundedlinearright-
inverse.ThisresultisneededtoestablishasemiflowforquasilinearproblemswithRobin
boundaryconditionsinChapter4.

IntheChapters4and5wethenapplyourlineartheorytoquasilinearreaction-diffusion
systemswithnonlinearboundaryconditions.Intentionallywedonotusethefullgenerality
ofthelineartheoryandratherfocusfromthebeginningonsomespecificproblemswhich
alsoallowforaninvestigationoftheirlong-timebehaviour.OnaboundeddomainΩwith

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smoothboundaryΓ=∂ΩandouterunitnormalfieldνweconsiderinChapter4systems
withRobinboundaryconditions,i.e,problemsoftheform
∂tu−∂i(aij(u)∂ju)=f(u)inΩ,t>0,
aij(u)νi∂ju=g(u)onΓ,t>0,(4)
u(0,∙)=u0inΩ.
Itisassumedthat(aij)isellipticandofseparateddivergenceform,andthatthenon-
linearitiesaresmooth.Adynamictheoryfor(4)inascaleofSlobodetskiispaceswas
establishedbyAmann[6]viaextrapolationtechniques.Localwell-posednessandinvariant
manifoldsnearequilibriafor(4)basedontheunweightedmaximalLp-regularityapproach
wereobtainedbyLatushkin,Prüss&Schnaubelt[65,66].
Ourfocusliesonthegloballong-timebehaviourinstrongnormsclosetoWp2,wherep<∞
isarbitrarilylarge.WeemploymaximalLp,µ-regularitytogetherwithregularityresultson
thesuperpositionoperatorsinducedbythenonlinearitiestoconstructinTheorem4.3.6a
compactlocalsemiflowofsolutionsfor(4)inthescaleofnonlinearphasespaces
Mps:=u0∈Wps(Ω,RN):aij(u0)νi∂ju0=g(u0)onΓ,
wherep∈(n+2,∞)ands∈(1+n/p,2−2/p].Thishighrangeofregularityisnotcovered
byAmann’stheory.InTheorem4.4.2wethenshowthataglobalattractorof(4)inMp2−2/p
existsifthereisanabsorbantsetinaHölderspaceCα(Ω,RN)forsomeα>0.Theresult
requiresthefullstrengthofthemaximalLp,µ-regularityapproachforthelinearizationof
(4)andpreciseestimatesforthenonlinearterms,whichcanbecontrolledintermsof
lowernormsofthesolution(seeLemma4.2.3).InparticularweobtainfromSobolev’s
embeddingsthatthesolutionsconvergetotheattractorwithrespecttotheC1+β(Ω,RN)-
norm,whereβ∈(0,1).Inimportantspecialcasesitsufficestohaveanabsorbantsetin
aweakernormsuchasthesup-norm.Improvingearlierresults,wethushaveestablished
thatthelong-timebehaviourofalsothespatialgradientofasolutionisdeterminedby
thedynamicsontheattractorwithrespecttoasup-norm.Theconvergenceinahigher
normcanbeusefultoimproveerrorestimatesfornumericalalgorithmswhenassumingin
aquasi-stationaryapproximationthatpartsofasystemofpartialdifferentialequations
areonafasttimescale.
TheabovestatementsaboutconvergenceinanormclosetoWp2areknownforsemilinear
problemswithlinearboundaryconditions,butdonotseemtoexistforquasilinearproblems
ornonlinearboundaryconditions.Relatedresultsrelyonthevariationofconstantsformula.
Theflexibilityoftheweightsbuildsabridgefromlowertohigherregularities,andthus
maximalLp,µ-regularitycanbeseenasasubstituteinthecaseofquasilinearproblems.
InSection4.5weapplyourresultstoshowconvergencetoanattractorinhighernorms
forconcretemodels.Weconsidersemilinearreaction-diffusionsystemswithnonlinear
boundaryconditions,asstudiedbyCarvalho,Oliva,Pereira&Rodriguez-Bernal[15],a
chemotaxismodelwithvolumefillingeffect,introducedbyHillen&Painter[53],andthe
Shigesada-Kawasaki-Teramotocross-diffusionmodelforpopulationdynamics,introduced
[76].in

tioncduIntro

7

InChapter5weturntosystemswithnonlineardynamicboundaryconditions,i.e.,prob-
formtheoflems∂tu=∂i(a1(u)∂iu)+a2(u)u+f(u),inΩ,t>0,
∂tu+b(∙,u)∂νu=divΓ(c1(∙,u)Γu)+c2(∙,u)Γu+g(∙,u),onΓ,t>0,(5)
u(0,∙)=u0,inΩ.
HereΩandthecoefficientsareasabove,andΓanddivΓdenotethesurfacegradientand
thesurfacedivergenceontheboundaryΓ,respectively.Althoughtheylookmorenonlinear
atafirstglance,suchboundaryconditionsareinfactlessnonlinearthantheonesin(4).
Infact,theautonomousversionoftheirlinearizationmaybecastintheabstractform(1)
byconsideringitasanevolutionequationontheproductspace
(v,vΓ)∈Lp(Ω,RN)×Wp1−1/p(Γ,RN):trΩv=vΓ,
wheretrΩdenotesthespatialtraceonΩ,andonemayidentifytheunknownuwiththe
pair(u,trΩu).Consequentlyonecanworkinlinearphasespacesevenforinitialregularities
closetoWp2.
Thesystem(5)modelsthebehaviourofthequantitiesundergoingareaction-diffusion-
convectionprocessinadomainandonitsboundary,coupledbythenormalfluxterm
b(∙,u)∂νu.Forb≡1,in[43,Section4]theeffectofthiscouplingisinterpretedassending
concentrationwavesfromΓintoaninfinitesimallayerneartheboundary.Similardynami-
calboundaryconditionsariseinCahn-HilliardorCaginalpphasefieldmodelsifonetakes
intoaccounttheshort-rangedinteractionwithwalls[73].Theyalsoariseintwophaseflows
withsolublesurfactant[14].Intheliteraturetheseboundaryconditionsarealsocalledgen-
eralizedWentzellboundaryconditions[43].Semilinearversionsof(5)withasingleequation
wereinvestigatedbymanyresearchers,forinstancebyFavini,J.A.Goldstein,G.R.Gold-
stein&Romanelli[38,39,40],Sprekels&Wu[80]andVazquez&Vitillaro[83].Resultson
quasilinearversionsdonotseemtoexist.Therearefurtherresultsonquasilinearsystems
withdynamicboundaryconditionsofreactivetype,i.e.,wheretangentialderivativesdo
notoccur.AdynamictheoryforsuchproblemswasestablishedbyEscher[36],basedon
Amann’swork.WerefertoConstantin&Escher[20]andthereferencesthereinformore
ts.elopmendevtrecen

Wefirstinvestigatethelinearinhomogeneous,nonautonomousversionof(5).Underappro-
priateellipticityconditionsonthecoefficientsmaximalLp,µ-regularityisshowninTheorem
5.2.1.Thisextendsthelinearresultsof[38,39,40,83]tomoregeneralproblemsandto
theLp-case.NextweconstructinTheorem5.3.3acompactlocalsemiflowofsolutionsfor
(5)inthelinearphasespace
M:=(v,vΓ)∈Wp2−2/p(Ω,RN)×Wp3−3/p(Γ,RN):trΩv=vΓ,
usingthelineartheoryandemployingtheideasandresultsof[59]onabstractquasilinear
evolutionequationsinLp,µ-spaces.AssuminganaprioriHölderbound,weareabletoshow
globalexistenceforasolutionof(5)inTheorem5.4.1.Hereagainmaximalregularity,

8

ionuctdIntro

localizationtechniquesinspaceandtimeandappropriatenonlinearestimatesarethe
crucialingredients.InSection5.5wespecializetoasemilinearversionof(5),
∂tu=Δu+f(u)inΩ,t>0,
∂tu+∂νu=ΔΓu+g(u)onΓ,t>0,(6)
u(0,∙)=u0inΩ,
andinvestigatethelong-timebehaviourofsolutionsintermsofglobalattractors.HereΔΓ
denotestheLaplace-Beltramioperatorontheboundary.Underappropriatedissipativityas-
sumptionsonthereactiontermsfandgweshowthatthesystem(6)possessesaLyapunov
function,andthatsolutionsareboundedintheenergyspaceW21(Ω,RN)×W21(Γ,RN).
EmployingaMoser-Alikakositerationprocedure,wethendeduceglobalexistence.Another
aprioriestimateontheequilibriaof(6)yieldstheexistenceofaconnectedglobalattractor,
andthateachsolutionconvergestothesetofequilibria(seeTheorem5.5.8).
Forasingleequationitisshownin[80]thateachsolutionof(6)convergestoanequilibrium
astimetendstoinfinity.Ourdissipativityconditionsdiffersfromtheonein[80],andis
rathercomparablewiththeonein[15]forRobinboundaryconditions.Wemayallowfor
onecomponentofareactiontermtohaveanunfavourablesign,providedthecorresponding
componentoftheotherreactiontermcompensatesthisappropriatelyintermsofpositivity
ofaRayleighquotientrelatedto(6).

Finally,intheappendixweprovidefactsfrominterpolationtheory,thetheoryofsectorial
operators,differentialoperatorsonaboundaryandaboutfunctionsspacesthatareused
throughoutthethesis.Wegiveprecisereferencesandalsoprovesome(rathersimple)
resultsforwhichareferencedoesnotseemtoexist.
Notations.WewriteR+n:={(x1,...,xn)∈Rn:xn>0}forn∈N,andC+:={z∈C:
Rez>0}.Ifitholdsa≤Cbfornonnegativequantitiesa,bwithagenericconstantC>0
wewriteab.TheLebesgue,SobolevandSlobodetskiispacesaredenotedbyLp,Hps
andWps,wherep∈[1,∞]designatesintegrabilityands∈Rdesignatesdifferentiability.
Forθ∈(0,1)andp∈[1,∞]wedenoteby(∙,∙)θ,pand[∙,∙]θtherealandthecomplex
interpolationfunctor,respectively.Thespaceofboundedlinearoperatorsbetweentwo
BanachspacesE,FisdenotedbyB(E,F),whereB(E):=B(E,E).IfFisdenselyand
continuouslyembeddedintoEwewriteF→dE,andifE,Fcoincideassetsandhave
equivalentnormswewriteE=F.Thedomain,thespectrumandtheresolventsetofa
closedoperatorAonEaredenotedbyD(A),σ(A)andρ(A),respectively.Forp∈[1,∞]
ands=[s]+s∗with[s]∈N0ands∗∈[0,1)wesetDA(s,p):=D(As)ifs∈N0and
DA(s,p):={x∈D(A[s]):A[s]x∈(E,D(A))s∗,p}fors∈/N0.
Acknowledgements.IwouldliketoexpressmygratitudetomysupervisorRoland
Schnaubeltforintroducingmetotheseinterestingtopicsandalwayshavinganopendoor
fordiscussions.FurtherIwouldliketothankhimforgivingmetheopportunitytopar-
ticipateinmanyconferencesandschools.Ialsobenefitedfromthefactthatthethesis
waspartiallyfinanciallysupportedbytheDeutscheForschungsgemeinschaft.Foradvices,

Introdutionc9

encouragementandsharingideasIamgratefultoRobertDenk,CiprianG.Gal,YuriLa-

tushkin,LahcenManiar,JanPrüss,GieriSimonett,LutzWeis,MathiasWilkeandRico

Zacher.SpecialthanksgotoAndreasBolleyer,PeerChristianKunstmannandAlexander

Ullmannforthepermanentdiscussionofdetails.Sincerethanksare

forsupportingmeineveryaspect

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for

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throughout

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thank

1Chapter

TheSpacesLp,µandWeighted
SpacesAnisotropic

Inthischapterweinvestigatethevector-valuedLp,µ-spacesandthecorrespondingweighted
anisotropicSobolev-Slobodetskiispacesinasystematicway,wherewerestricttospacesof
classHTfromthebeginning(cf.AppendixA.3).WefirstconsidertheSobolev-Slobodetskii
spacesoverthehalf-lineandafiniteinterval,andderivebasicproperties.Ofparticular
importanceisherethatthederivativewithpositiveandnegativesignadmitsabounded
H∞-calculusontheLp,µ-spacesoverthehalf-line,respectively.Nextwebrieflyreviewthe
resultsfrom[71]and[42]onabstractmaximalLp,µ-regularityandoperator-valuedFourier-
multipliersinLp,µ.Thenweturntoanisotropicspaces,andinvestigatetheNewtonpolygon,
temporalandspatialtracesandpointwisemultipliers.
Throughoutweusethefactsoninterpolationtheory,sectorialoperatorsandfunctionspaces
reviewedintheappendix.

1.1ertiesPropBasic

Let(E,|∙|E)beacomplexBanachspaceofclassHTandletJ=R+=(0,∞)orJ=(0,T)
forsomeT>0.Letfurther
p∈(1,∞),µ∈(1/p,1].
Foru:J→Ewedenotebyt1−µuthefunctiont→t1−µu(t)onJ.Wedefine
Lp,µ(J;E):=u:J→E:t1−µu∈Lp(J;E),
whichbecomesaBanachspacewhenequippedwiththenorm
/p1|u|Lp,µ(J;E):=|t1−µu|Lp(J;E)=tp(1−µ)|u(t)|Epdt.
JNotethatµ=1correspondstotheunweightedcase,Lp,1=Lp,andthattheweighttp(1−µ)
onlyhasaneffectatt=0andt=∞.Thus,forT>0,
Lp(0,T;E)→Lp,µ(0,T;E),Lp,µ(0,T;E)→Lp(τ,T;E),τ∈(0,T),

12

TheSpacesLp,µandWeightedAnisotropicSpaces

butLp(R+;E)Lp,µ(R+;E)forµ∈(1/p,1).Fork∈N0wedefinethecorresponding
weightedSobolevspace
Wkp,µ(J;E)=Hkp,µ(J;E):=u∈W1k,loc(J;E):u(j)∈Lp,µ(J;E),j∈{0,...,k},
whichbecomesaBanachspacewhenequippedwiththenorm
kp1/p
|u|Wkp,µ(J;E)=|u|Hkp,µ(J;E):=|u(j)|Lp,µ(J;E).
=0jFors∈R+\Nwiths=[s]+s∗,where[s]∈N0,s∗∈(0,1),wedefineweightedSlobodetskii
andBesselpotentialspacesbyrealandcomplexinterpolation,respectively,i.e.,
Wsp,µ(J;E):=W[p,µs](J;E),W[p,µs]+1(J;E)s∗,p,
Hsp,µ(J;E):=W[p,µs](J;E),W[p,µs]+1(J;E)s∗.
ByPropositionA.4.2thisdefinitionisconsistentwiththeunweightedcase,i.e.,wehave
Wps=Wp,s1andHsp=Hp,s1foralls≥0.Thegeneralpropertiesofrealandcomplex
interpolationspaces(AppendixA.2)implythatforfixedp∈(1,∞)andµ∈(1/p,1]one
scalethehasWsp,µ1→dHsp,µ2→dWsp,µ3→dHsp,µ4,s1>s2>s3>s4≥0.(1.1.1)
Inthesequelwewilloftenusethat
B(Wkp,µ(J;E))∩B(Wkp,µ+1(J;E))→B(Wsp,µ(J;E))∩B(Hsp,µ(J;E)),
wherek∈N0ands∈(k,k+1),whichmeansthatitsufficestoconsiderthespacesof
integerordertoshowthatonoperatoriscontinuousontheWsp,µ-andHsp,µ-scale.
Beforecontinuingwithdefinitions,wederiveafirstbasicpropertyofLp,µ.
Lemma1.1.1.LetJ=(0,T)beafiniteorinfiniteinterval,p∈(1,∞)andµ∈(1/p,1].
Then1Lp,µ(J;E)→Lq,loc(J;E),1≤q<1−µ+1/p.
Consequently,fork∈NitholdsWkp,µ(J;E)→W1k,loc(J;E),andforu∈Wkp,µ(J;E)the
traceu(j)(0)∈Eexistsforj∈{0,...,k−1}.
Proof.ForafiniteintervalJ=(0,T)⊂J,afunctionu∈Lp,µ(J;E)and1≤q<p,
yieldsyinequalitHölder’sTTT(1−µ)pqpp−q
|u(t)|qEdt=t−q(1−µ)(t1−µ|u(t)|E)qdt≤t−p−q|u|Lqp,µ(J;E),
000wheretheintegralontheright-handsideisboundedfor(1p−−µq)pq<1,i.e.,q<1−µ1+1/p.

rtiesePropBasic1.1

13

InviewofLemma1.1.1itmakessensetodefine
0Wkp,µ(J;E)=0Hkp,µ(J;E):=u∈Wkp,µ(J;E):u(j)(0)=0,j∈{0,...,k−1}
fork∈N,andforconveniencewefurtherset
0W0p,µ(J;E)=0H0p,µ(J;E):=Lp,µ(J;E).
Foranumbers=[s]+s∗∈R+\Nasaboveweagaindefinethecorrespondingfractional
orderspacesbyinterpolation,i.e.,
0Wsp,µ(J;E):=0W[p,µs](J;E),0W[p,µs]+1(J;E)s∗,p,
0Hsp,µ(J;E):=0W[p,µs](J;E),0W[p,µs]+1(J;E)s∗.
Thisyieldsasbeforeascaleoffunctionspaces
0Wsp,µ1→d0Hsp,µ2→d0Wsp,µ3→d0Hsp,µ4,s1>s2>s3>s4≥0,(1.1.2)
andwefurtherhavethat
0Wsp,µ(J;E)→Wsp,µ(J;E),0Hsp,µ(J;E)→Hsp,µ(J;E),s>0.
ThefollowingfundamentalHardyinequalitiesareavailableforthespacesbasedonvan-
alues.vinitialishingLemma1.1.2.LetJ=(0,T)befiniteorinfiniteandp∈(1,∞).Thenthefollowing
true.holdsa)Forα∈(1/p,∞)andanonnegativefunctionu∈L1,loc(R+;E)itholds
∞tp1∞
0t−α0u(τ)dτdt≤(α−1/p)p0(t1−α|u(t)|)pdt.
b)Forµ∈(1/p,1]andk∈N0itholds
tp(1−µ−k)|u(t)|Epdt≤Cp,µ,k|u(k)|Lpp,µ(J;E)ifu∈0Wkp,µ(J;E).
Jc)Forµ∈(1/p,1]ands≥0itholds
tp(1−µ−s)|u(t)|Epdt≤Cp,µ,s|u|p0Wsp,µ(J;E)ifu∈0Wsp,µ(J;E),
Jfurtherandtp(1−µ−s)|u(t)|Epdt≤Cp,µ,s|u|p0Hsp,µ(J;E)ifu∈0Hsp,µ(J;E).
J

14

TheSpacesLp,µandWeightedAnisotropicSpaces

Proof.Theestimateina)isshownasin[50,Theorem330].Forb),inthesequelwe
identifyu∈0Wkp,µ(J;E)anditsderivativeswiththeirtrivialextensionstoR+.Thenfor
j∈{1,...,k}wehaveu(j)∈L1,loc([0,∞);E),and,sinceu(j−1)(0)=0,itholds
t|u(j−1)(t)|E≤|u(j)(τ)|Edτ,t∈R+.
0Forα>1/pitthusfollowsfroma)that
∞tp
t−pα|u(j−1)(t)|Epdt≤t−α|u(j)(τ)|Edτdt
00J≤1pt−p(α−1)|u(j)(t)|pdt.
(α−1/p)J
Applyingthisinequalityktimes,withαj=µ+k−j>1/pforj∈{1,...,k},weobtain
theassertedestimateinb).Toprovec),wesetfors≥0
Lp(J,tp(1−µ−s)dt;E):=u:J→E:t1−µ−su∈Lp(J;E).
Itthenfollowsfromb)that
0Wkp,µ(J;E)→Lp(J,tp(1−µ−k)dt;E),k∈N0.(1.1.3)
In[82,Theorem1.18.5]theidentity
(Lp(J,tp(1−µ−k)dt;E),Lp(J,tp(1−µ−(k+1))dt;E))θ,p=Lp(J,tp(1−µ−θk)dt;E)(1.1.4)
isshown,wherek∈N0andθ∈(0,1),and(1.1.4)remainstrueifonereplaces(∙,∙)θ,pby
thecomplexinterpolationfunctor[∙,∙]θ.Hencec)followsfrom(1.1.3)byinterpolation.
WeusetheHardyinequalitiestoshowthatthemultiplicationwiththeweightisaniso-
morphismtotheunweightedspaces,providedonerestrictstovanishinginitialvalues.The
followingresultisshownin[71,Proposition2.2]fors=0ands=1.
Lemma1.1.3.LetJ=(0,T)befiniteorinfinite,p∈(1,∞),µ∈(1/p,1]ands≥0.
ThenthemapΦµ,givenby
(Φµu)(t):=t1−µu(t),
inducesanisomorphismbetween0Wsp,µ(J;E)and0Wps(J;E),andbetween0Hsp,µ(J;E)
and0Hps(J;E).TheinverseΦµ−1isgivenby(Φµ−1u)(t)=t−(1−µ)u(t).
Proof.Byinterpolationweonlyhavetoconsiderthecases=k∈N0.
(I)ClearlyΦµisanisomorphismincasek=0.Fork∈Nwetakeu∈0Wkp,µ(J;E)and
estimateforj∈{1,...,k},usingLemma1.1.2,
j|(t1−µu)(j)|Lpp(J;E)t−p(µ+i−1)|u(j−i)(t)|Epdt|u|pWj(J;E).
p,µJ=0i

rtiesePropBasic1.1

15

ThusΦµmaps0Wkp,µ(J;E)continuouslyintoWpk(J;E).Sinceu∈Ck−1(J;E),itholdsfor
j∈{0,...,k−1}
jj|(t1−µu)(j)(t)|Et1−µ−i|u(j−i)(t)|Et−i|u(j−i)(t)|→0,t0,
=0i=0iandthisshows(Φµu)(j)(0)=0forallj∈{0,...,k−1}.
(II)Nowtakeu∈0Wpk(J;E)andj∈{1,...,k}.Then,againbyLemma1.1.2,
j|(t−(1−µ)u)(j)|Lpp,µ(J;E)t−pi|u(j−i)|Epdt|u(j)|Lpp(J;E),
J=0iwhichyieldsthatΦµ−1maps0Wpk(J;E)continuouslyintoWkp,µ(J;E).Moreover,forj∈
{0,...,k−1}itholds
j|(Φµ−1u)(j)(t)|Etµ−1−i|u(j−i)(t)|Etµ−1sup|u(j)(τ)|E
ti=0τ∈(0,t)
≤tµ−1|u(j+1)(τ)|Edτtµ−1/p|u(j+1)|Lp(J;E),
0whichconvergestozeroast0.Hence(Φµ−1u)(j)(0)=0forallj∈{0,...,k−1}.
Wenextshowbasicdensityresultsfortheweightedspaces.
Lemma1.1.4.ForafiniteorinfiniteintervalJ=(0,T),p∈(1,∞),µ∈(1/p,1]and
holdsit0s≥Cc∞(J\{0};E)→d0Wsp,µ(J;E),0Hsp,µ(J;E),Cc∞(J;E)→dWsp,µ(J;E),Hsp,µ(J;E).
Proof.Bythegeneraldensityresultsforinterpolationspaces(AppendixA.2),weonly
havetoconsiderthecases=k∈N0.Throughout,letε>0begiven.
(I)Foru∈0Wkp,µ(J;E)itholdsΦµu∈0Wpk(J;E)bytheprecedinglemma.Asin[82,
Theorems2.9.1,4.7.1]forthescalar-valuedcase,oneseesthatCc∞(J\{0};E)isdensein
0Wpk(J;E)fork∈N0.Thusthereisψ∈Cc∞(J\{0};E)with|Φµu−ψ|Wpk(J;E)<ε.
Therefore|u−Φµ−1ψ|Wkp,µ(J;E)|Φµu−ψ|Wpk(J;E)ε,
whichyieldsCc∞(J\{0};E)→d0Wkp,µ(J;E).
(II)Toshowthesecondasserteddensity,takeu∈Wkp,µ(J;E)andchooseψ1∈Cc∞(J;E)
withψ1(j)(0)=u(j)(0)forj∈{0,...,k−1}.1ByStepI,duetou−ψ1∈0Wkp,µ(J;E),there
isψ2∈Cc∞(J;E)whichisε-closetou−ψ1.Henceψ1+ψ2isε-closetou.
ForafiniteintervalJ=(0,T),alinearmapE:L1,loc(J;E)→L1,loc(R+;E)iscalled
extensionoperatorfromJtoR+if
(Eu)|J=u,u∈L1,loc(J;E),
1Forinstance,onemaytakeψ1(t)=ϕ(t)Pjk=0j1!u(j)(0)tj,whereϕ∈Cc∞([0,∞)withϕ≡1on[0,1]
andϕ≡0on[2,∞).

16

TheSpacesLp,µandWeightedAnisotropicSpaces

i.e.,ifitisaright-inversetotherestrictionoffunctionsonR+toJ.
Weconstructextensionoperatorsfortheweightedspaces.Inthesequeltheyarefrequently
employedtodeducepropertiesoftheweightedspacesonafiniteintervalfromthehalf-
linecase.Fors∈[0,2]weconstructonefor0Wsp,µ(J;E)and0Hsp,µ(J;E)whosenormis
independentofthelengthofJ.Wedonotconsidersuchanextensionfors>2,sincethis
caseisnotneededbelowforourlaterapplicationsandtheconstructionwouldberather
ersome.bcumLemma1.1.5.LetJ=(0,T)beafiniteinterval,p∈(1,∞),andµ∈(1/p,1].Thenthe
true.dsholwingfolloa)Givenk∈N,thereisanextensionoperatorEJfromJtoR+with
EJ∈B(Wsp,µ(J;E),Wsp,µ(R+;E))∩B(Hsp,µ(J;E),Hsp,µ(R+;E)),s∈[0,k].
b)ThereisanextensionoperatorEJ0fromJtoR+with
EJ0∈B(0Wsp,µ(J;E),0Wsp,µ(R+;E))∩B(0Hsp,µ(J;E),0Hsp,µ(R+;E)),s∈[0,2],
whoseoperatornormisindependentofT.
c)FortheaboveoperatorsitholdsEJ,EJ0∈B(L∞(J;E),L∞(R+;E)),withoperator
normsindependentofT.
Proof.(I)ForEJ,letk∈Nbegiven.By[1,Theorem5.19]thereisanextensionoperator
EfromR+toRthatiscontinuousfromWpj(R+;E)toWpj(R;E)forallj∈{0,...,k},that
satisfies(Ev)(j)=Ejv(j),j∈{0,...,k},(1.1.5)
whereEjisanextensionoperatorthatiscontinuousfromWpi(R+;E)toWpi(R;E)for
alli∈{0,...,k−j}.FurtherEandEjhavethepropertythatforT>0thefunction
E(j)v|(−Te/(k+1),0)isconstructedusingonlyv|(0,Te).Wemaythusdefine
(Eu)(t):=Eu(−∙+T)(−t+T),t∈0,T+T/2(k+1),u∈L1,loc(0,T;E).
ThenEisanextensionoperatorfrom(0,T)to0,(1+2(k1+1))T.Dueto(1.1.5),and
sincetheweightonlyhasaneffectatt=0,forallj∈{0,...,k}itiscontinuousfrom
Wjp,µ(0,T;E)toWjp,µ0,(1+4(k1+1))T;E.Choosingacut-offfunctionϕ∈Cc∞(R+)that
isequalto1on0,(1+4(k1+1))Tandvanisheson(1+3(k1+1))T,∞,wedefine
EJ:=ϕE.(1.1.6)
ThenitholdsEJ∈B(Wjp,µ(J;E),Wjp,µ(R+;E))forj∈{0,...,k},whichcarriesovertothe
fractionalorderspacesbyinterpolation.ItfollowsfromtherepresentationofEin[1]that
EJadmitsanL∞-estimateindependentofT.
(II)Toshowb),foru∈0Wkp,µ(J;E)wedefine
0u(t),t∈(0,T),
(EJu)(t):=3(ψ1−µu)(2T−t)1[T,2T](t)−2(ψ1−µu)(3T−2t)1[T,3T](t),t≥T,
2

1.1rtiesePropBasic

17

2whereψ(τ)=2TTτ2−τ.Asabove,byinterpolationweonlyhavetoshowthatEJ0∈
B(0Wkp,µ(J;E),0Wkp,µ(R+;E))fork=0,1,2.Fork∈{1,2}weseethatthefunction
EJ0uiscontinuouson[0,∞).Further,inthesecasesitholds
(EJ0u)(t)=−3(ψ1−µu)(2T−t)1[T,2T](t)+4(ψ1−µu)(3T−2t)1[T,3T](t),t≥T,
2andforτ∈Jwehave
1)−µ2((ψ1−µu)(τ)=2(1−µ)T(T−τ)τ−µu(τ)+T2(µ−1)(2T−τ)1−µτ1−µu(τ).(1.1.7)
(2T−τ)µ
Fork=2wethusobtainthatlimtT(EJ0u)(t)=u(T),andweinferfromu(0)=0and
|u(τ)|Eτ−µ|u(τ)|Eτ−1→|u(0)|E=0,τ0,
that(EJ0u)iscontinuousatt=23Tandt=2T.Therefore(EJ0u)iscontinuouson[0,∞).
Moreover,inthiscaseitholdsforτ∈Jthat
(T−τ)2T2(µ−1)
(ψ1−µu)(τ)=4(1−µ)−µT2(µ−1)τ−µ−1−τ−µu(τ)
(2T−τ)1+µ(2T−τ)µ
τT−+4(1−µ)T2(µ−1)τ−µu(τ)+T2(µ−1)(2T−τ)1−µτ1−µu(τ).(1.1.8)
(2T−τ)µ
(III)WeestimateEJ0uanditsderivativesintheweightednorms.Usingforj=1,2the
substitutionsτ=(j+1)T−jt(i.e.,t=(j+1)jT−τ),wehave
+1j2T0ppjp(1−µ)1−µp
|EJu|Lp,µ(R+;E)|u|Lp,µ(J;E)+t|(ψu)((j+1)T−jt)|Edt
T=1j|u|p+((j+1)T−τ)(2T−τ)τp(1−µ)|u(τ)|pdτ
2Tp(1−µ)
Lp,µ(J;E)0T2E
=1jp|u|Lp,µ(J;E),
whichyieldsEJ0∈B(Lp,µ(J;E),Lp,µ(R+;E)),withoperatornormindependentofT.Sim-
ilarly,foru∈0W1p,µ(J;E)weobtain,using(1.1.7)andHardy’sinequality(Lemma1.1.2),
20ppT(j+1−τ/T)p(1−µ)(T−τ)p−pµp
|(EJu)|Lp,µ(R+;E)|u|Lp,µ(J;E)+Tp(1−µ)(2T−τ)pµτ|u(τ)|E
0=1j+(2T−τ)p(1−µ)τp(1−µ)|u(τ)|pdt
Ep|u|Lp,µ(J;E).
Moreover,foru∈0W2p,µ(J;E)weuse(1.1.8)andHardy’sinequalitytoestimate
2T0ppp(1−µ)2p(µ−1)
|(EJu)|Lp,µ(R+;E)|u|Lp,µ(J;E)+((j+1)T−τ)T
0=1jpp2∙(T−τ)τ−p(µ+1)+Tτ−p(µ+1)|u(τ)|p
(2T−τ)p(1+µ)(2T−τ)pµE
(T−τ)p
+τ−pµ|u(τ)|p+(2T−τ)p(1−µ)τp(1−µ)|u(τ)|pdτ
(2T−τ)pµE
p|u|Lp,µ(J;E),

18

TheSpacesLp,µandWeightedAnisotropicSpaces

wheretheconstantsintheseestimatesareindependentofT.ThisshowsthatEJ0∈
B(0Wkp,µ(J;E),0Wkp,µ(R+;E))fork=1,2,withoperatornormindependentofT,respec-
tively.Finally,itfollowsagainfromitsrepresentationthatEJ0admitsanL∞-estimate
independentofT.
Wenowinvestigatetherealizationofthederivative∂tanditsfractionalpowersonthe
weightedspaces.Thepropertiesofthisoperatoranditsvariantsarefundamentalforall
ourfurtherconsiderations.Wefirstshowthat∂tgeneratesthefamilyoflefttranslations.
Lemma1.1.6.Forp∈(1,∞),µ∈(1/p,1]ands≥0,thefamilyoflefttranslations
{ΛtE}t≥0,givenby
(ΛtEu)(τ):=u(τ+t),τ≥0,
iswell-definedandformsastronglycontinuoussemigroupofcontractionsonthespaces
Wsp,µ(R+;E)andHsp,µ(R+;E),respectively.Itsgeneratoristhederivative∂t,withdomain
Wsp,µ+1(R+;E)andHsp,µ+1(R+;E),respectively.
Proof.(I)WewriteΛt=ΛtEforsimplicity.Foreacht0≥0theoperatorΛt0maps
Lp,µ(R+;E)intoitselfandiscontractive,dueto
∞|Λt0u|Lpp,µ(R+;E)=τp(1−µ)|u(τ+t0)|Epdτ
0∞≤(τ+t0)p(1−µ)|u(τ+t0)|Epdτ≤|u|Lpp,µ(R+;E).
0ThisestimatealsoshowsthatΛt0mapsWkp,µ(R+;E),k∈N,intoitselfandiscontractive.
Byinterpolation,thiscarriesovertoWsp,µ(R+;E)andHsp,µ(R+;E),foralls≥0.Itis
furtherclearthat{Λt}t≥0formsasemigroupofoperatorsonthesespaces.DuetoLemma
1.1.4,thesetCc∞([0,∞);E)isdenseinallofthespacesabove,andthelefttranslations
actstronglycontinuousonthisset.By[35,PropositionI.5.3],thisyieldsthattheleft
translationsarestronglycontinuousonWsp,µ(R+;E)andHsp,µ(R+;E),respectively.
(II)Nowletk∈N0.Denotingthegeneratorof{Λt}onWkp,µ(R+;E)byA,wehaveto
showthat∂t=A.ToseeA⊆∂t,wetakeu∈D(A).Thenu(k)∈L1,loc(R+;E),andfor
a,b∈R+witha<bitholds
b1u(k)(τ+h)−u(k)(τ)dτ=1b+hu(k)(τ)dτ−1a+hu(k)(τ)dτ.
ahhbha
Ash→0,theleft-handsideconvergestou(k)(b)−u(k)(a)foralmostalla,b∈R+.
TheintegrandontherightconvergestoAu(k)inLp,µ(R+;E),andthusinL1(a,b;E).
Hence,theright-handsideconvergestoabAu(k)(τ)dτ.Thisshowsu∈W1k,lo+1c(R+;E),
withu(k+1)=Au(k).ThusD(A)⊂Wkp,µ+1(R+;E)and∂t|D(A)=A.
Thereverseinclusionnowfollowsfromabstractarguments.SinceAgeneratesastrongly
continuoussemigroupofcontractions,itfollowsfromtheHille-Yosidatheorem,[35,The-
oremII.3.5],that1−Aisinvertible.Itisfurthereasytoseethat1−∂tisinjectiveon
Wkp,µ+1(R+;E).From[35,IV.1.21(5)]wethusobtainthat1−∂t=1−A,whichyields
.A=∂t

rtiesePropBasic1.1

19

(III)Fors≥0weconcentrateontheW-case,theH-caserequiresliterallythesame
arguments.By[35,PropositionII.2.3]wehavethatthegeneratorofthelefttranslations
onWsp,µ(R+;E)isthederivative∂taswell,withdomain
D(∂t)={u∈Wsp,µ(R+;E):∂tu∈Wsp,µ(R+;E)}.
ItfollowsfrominterpolationthatWsp,µ+1(R+;E)⊂D(∂t).Fortheconverseinclusion,if
u,∂tu∈Wsp,µ(R+;E)then(1−∂t)u∈Wsp,µ(R+;E).Since1iscontainedintheresolvent
setof∂t,StepIIandinterpolationyieldu∈Wsp,µ+1(R+;E).Therefore∂twithD(∂t)=
Wsp,µ+1(R+;E)isthegeneratorofthelefttranslationsalsointhefractionalordercase.
Usingatransferenceprinciple,weshowthatthenegativegeneratorofthelefttranslations,
−∂t,admitsaboundedH∞-calculusonLp,µ(R+;E).Asshownin[71],therealizationof
∂twithdomain0W1p,µ(R+;E)alsoadmitsaboundedH∞-calculus,although−∂tdoesnot
generateasemigrouponLp,µ(R+;E).ForadefinitionandpropertiesoftheH∞-calculus
ofasectorialoperatorwerefertoAppendixA.3.
Theorem1.1.7.2Letp∈(1,∞)andµ∈(1/p,1].ThenonLp,µ(R+;E)theoperators
∂t,withdomain0W1p,µ(R+;E),
and−∂t,withdomainW1p,µ(R+;E),
admitaboundedH∞-calculuswithH∞-angleπ/2,respectively.Inparticular,bothoper-
atorsaresectorialofangleπ/2.
Proof.(I)Theassertionon∂tisprovedin[71,Theorem4.5].
(II)Fortheoperator−∂tweemploythevector-valuedtransferenceprinciple,whichisdue
toHieberandPrüss[52].Tothisendweintroducevector-valuedextensionsofoperators.
Let(Ω,ν)beameasurespace,andletSbeabounded,positiveoperatoronLp(Ω,ν).3Let
furtherui∈Lp(Ω,ν)bestepfunctionsandxi∈E,i=1,...,N,whereN∈N.Forsimple
functionsuoftheformN
u=uixi,(1.1.9)
=1ithevector-valuedextensionofS,denotedbySE,isdefinedas
NSEu(∙):=(Sui)(∙)xi.
=1iDueto[62,Lemma10.14],theoperatorSEextendsuniquelytothevector-valuedspace
Lp(Ω,ν;E),suchthatSEB(Lp(Ω,ν;E))=SB(Lp(Ω,ν)).
(III)WeconsideronLp,µ(R+;E)=Lp(R+,tp(1−µ)dt;E)thelefttranslationΛtE,t≥0.
ObviouslyΛtRisapositiveoperator,andforasimplefunctionuoftype(1.1.9)itholds
ΛtEu=(ΛtR)Eu.
23TheHereopitiseratorforStheisfirstcalledtimepositivesseneitialfitthatleavEesistheofpclassositiveHT.cone{u∈Lp(Ω,ν):u≥0ν-a.e.}invariant.

20

TheSpacesLp,µandWeightedAnisotropicSpaces

FromthedensityofthesimplefunctionsinLp(R+,tp(1−µ)dt;E)itfollowsthatΛtEisthe
vector-valuedextensionofΛtR,i.e.
ΛtE=(ΛtR)E.
DuetoLemma1.1.6,thefamily{ΛtR}t≥0formsastronglycontinuoussemigroupofpositive
contractionsonLp(R+,tp(1−µ)dt),and∂tisthegeneratorofitsvector-valuedextension
{ΛtE}t≥0toLp(R+,tp(1−µ);E).Moreover,∂tisinjectiveonLp(R+,tp(1−µ);E).Now[52,
Theorem6]yieldsthat−∂tadmitsaboundedH∞-calculuswithangleequaltoπ/2.
Theinvertibilityof1−∂tand1+∂tyieldsausefulcharacterizationoftheweightedspaces.
Lemma1.1.8.ForafiniteorinfiniteintervalJ=(0,T)ands=[s]+s∗with[s]∈N0,
s∗∈[0,1),itholds
Wsp,µ(J;E)={u∈W[p,µs](J;E):u([s])∈Wsp,µ∗(J;E)},(1.1.10)
0Wsp,µ(J;E)={u∈0W[p,µs](J;E):u([s])∈0Wsp,µ∗(J;E)},(1.1.11)
wherethespacesontheright-handsideareequippedwiththeircanonicalnorms.4The
normequivalenceconstantsin(1.1.11)doesnotdependonthelengthofJ.Allthese
assertionsremaintrueifonereplacestheW-spacesbytheH-spaces.
Proof.WeonlyconsiderthecaseofW-spaces.
(I)Itfollowsfrominterpolationthat
∂t[s]∈BWsp,µ(J;E),Wsp,µ∗(J;E)∩B0Wsp,µ(J;E),0Wsp,µ∗(J;E),
whichshowstheembeddingsfromthelefttotherightin(1.1.10)and(1.1.11),withem-
beddingconstantsindependentofthelengthofJ.
(II)FortheconverseembeddingwefirstconsiderthecaseJ=R+.Since−∂tissectorial
wehavethat1−∂tisinvertible.Further,interpolationyieldsthattheoperator(1−∂t)[s]
isanisomorphismWsp,µ(R+;E)→Wsp,µ∗(R+;E).Wemaythereforeestimate
|u|Wsp,µ(R+;E)|(1−∂t)[s]u|Wsp,µ∗(R+;E)|u|W[p,µs](R+;E)+|u([s])|Wsp,µ∗(R+;E),
andthusobtain(1.1.10).Replacing1−∂tby1+∂t,whichisinvertiblesince∂tissectorial,
weobtain(1.1.11)inthesameway.
(III)NowsupposethatJisafiniteinterval.UsingtheextensionoperatorEJfromLemma
1.1.5and(1.1.10)forthehalf-line,weobtain
|u|Wsp,µ(J;E)|EJu|W[sp,µ](R+;E)+|(EJu)([s])|Wsp,µ∗(R+;E)|u|W[p,µs](J;E)+|u([s])|Wsp,µ∗(J;E),
whichshows(1.1.10).Here,thelatterinequalityfollowsfromtherepresentation(1.1.6)of
EJ.For(1.1.11),notethattheoperator1+∂tisalsoforafiniteintervalanisomorphism
0W1p,µ(J;E)→Lp,µ(J;E).Infact,itisobviouslyinjective.Moreover,thesolutionof
u=−u+fwithu(0)=0isforf∈Lp,µ(R+;E)givenbyu=u|J,whereu∈0W1p,µ(R+;E)
´`4Forinstance,in(1.1.10)thecanonicalnormontheright-handsideis|u|pWsp,µ(J;E)+|u([s])|pWsp,µ∗(J;E)1/p.

rtiesePropBasic1.1

21

(1.1.12)

satisfiesu=−u+f,andfdenotesthetrivialextensionofftoR+.Thisshowssurjectivity,
andfurtherthattheoperatornormof(1+∂t)−1doesnotdependonthelengthofJ.Now
thesameargumentsasinStepIIshow(1.1.11)forfiniteJ.
Wenextshowgeneralinterpolationpropertiesoftheweightedspaces.
Lemma1.1.9.LetJ=(0,T)befiniteorinfinite,p∈(1,∞),µ∈(1/p,1],0≤s1<s2
andθ∈(0,1).Thenfors=(1−θ)s1+θs2itholds
[Hsp,µ1(J;E),Hsp,µ2(J;E)]θ=Hsp,µ(J;E),
andifs/∈Nthen
(Hsp,µ1(J;E),Hsp,µ2(J;E))θ,p=Wsp,µ(J;E).(1.1.12)
Moreover,fors1,s2,s∈/N0itholds
[Wsp,µ1(J;E),Wsp,µ2(J;E)]θ=Wsp,µ(J;E),
(Wsp,µ1(J;E),Wsp,µ2(J;E))θ,p=Wsp,µ(J;E).
IfF→dEisafurtherBanachspaceofclassHT,thenitholdsforτ≥0andθ∈(0,1)
(Hτp,µ(R+;E),Hτp,µ(R+;F))θ,p=∙Hτp,µ(R+;(E,F)θ,p),
[Hτp,µ(R+;E),Hτp,µ(R+;F)]θ=∙Hτp,µ(R+;[E,F]θ).
AlltheseassertionsremaintrueifonereplacestheW-andH-spacesby0W-and0H-
spaces,respectively.Restrictingtos2≤2inthiscase,thenormequivalenceconstantsare
independentoftheunderlyingintervalJ.
Proof.Throughoutthisproofweset
A:=1−∂t,X:=Lp,µ(R+;E).
(I)WefirsttreatthecaseJ=R+.ConsideringAasanoperatoronX,Theorem1.1.7,
(A.3.1)and(A.3.2)yieldthatforα∈(0,1)itholdsD(Aα)=Hαp,µ(R+;E).Usingthis,
togetherwiththefactthatD(Ak)=Hkp,µ(R+;E)fork∈N0andLemma1.1.8,forα≥1
thatobtainalsoewD(Aα)={u∈D(A[α]):A[α]u∈D(Aα−[α])}
={u∈H[p,µα](R+;E):u[α]∈Hαp,µ−[α](R+;E)}=Hαp,µ(R+;E).
Itthereforefollowsfrom(A.3.1)that
[Hsp,µ1(R+;E),Hsp,µ2(R+;E)]θ=[D(As1),D(As2)]θ=D(As)=Hsp,µ(R+;E),
whichshowsthefirstassertedequality.
(II)Wenextshow(1.1.12).TheoperatorAs1inducesanisomorphism
(Hsp,µ1(J;E),Hsp,µ2(J;E))θ,p=(D(As1),D(As2))θ,p→(X,D(Aτ))θ,p,

22

TheSpacesLp,µandWeightedAnisotropicSpaces

whereτ=s2−s1.Itfollowsfromreiterationthat
(X,D(Aτ))θ,p=(D(A[τ]),D(Aτ))σ,p,
withσ=ττθ−−[τ[τ]]∈(0,1).Further,theoperatorA[τ]inducesanisomorphism
(D(A[τ]),D(Aτ))σ,p→(X,D(Aτ−[τ]))σ,p=(X,D(A))σ(τ−[τ]),p=Wσp,µ(τ−[τ])(R+;E),
whereσ(τ−[τ])=τθ−[τ]∈(0,1).NowtheoperatorA−(s1+[τ])inducesanisomorphism
Wτp,µθ−[τ](R+;E)→W(p,µs2−s1)θ+s1(R+;E)=Wsp,µ(R+;E),
provideds∈/N.
(III)Forthethirdequalitywetakeanintegerk>s2,andusetheassumptions,s1,s2∈/N0,
(1.1.12),A.2h),thereflexivityofXandD(Ak)andagain(1.1.12),toobtain
[Wsp,µ1(R+;E),Wsp,µ2(R+;E)]θ=[(X,D(Ak))s1/k,p,(X,D(Ak))s2/k,p]θ
=(X,D(Ak))s/k,p=Wsp,µ(R+;E).
Similarargumentsyieldthefourthassertedequality,i.e.,
(Wsp,µ1(R+;E),Wsp,µ2(R+;E))θ,p=((X,D(Ak))s1/k,p,(X,D(Ak))s2/k,p)θ,p
=(X,D(Ak))s/k,p=Wsp,µ(R+;E).
(IV)NowletF→dEbeafurtherBanachspaceofclassHTandτ≥0.Thentheoperator
Aτisanisomorphism
(Hτp,µ(R+;E),Hτp,µ(R+;F))θ,p→(Lp,µ(R+;E),Lp,µ(R+;F))θ,p.
Dueto[82,Theorem1.18.4],thelatterspaceequalsLp,µ(R+;(E,F)θ,p),andA−τmaps
thisspaceisomorphicallytoHτp,µ(R+;(E,F)θ,p).Thecorrespondingassertiononcomplex
interpolationisshowninthesameway.
(V)ReplacingtheoperatorA=1−∂tbyA0:=1+∂t,thesameargumentsasabove
showtheassertedequalitiesforthe0W-andthe0H-spaces.ThisfinishesthecaseJ=R+.
Thecaseofafiniteintervalcanbededucedfromthehalf-linecase,usingtheextension
operatorsEJandEJ0fromLemma1.1.5.Forinstance,onedecomposestheidentityintoEJ
andtherestrictionRJtoJandobtains
|u|[Hsp,µ1(J;E),Hsp,µ2(J;E)]θ≤|EJu|[Hsp,µ1(R+;E),Hsp,µ2(R+;E)]θ|EJu|Hsp,µ(R+;E)|u|Hsp,µ(J;E).
Theconverseembeddingisderivedinthesameway.5Thedependenceofthenormequiv-
alenceconstantsonthelengthofJcarriesoverfromthepropertiesoftheextensionoper-
ators.Notethathereitisimportantthattheextensionoperatorsactonawholescaleof
W-andH-spaces.
Thefollowingresultshowsthatthegoodpropertiesof1−∂tand1+∂tcarryovertothe
wholeW-andH-scale.
5Thisisnothingbuttheretraction-coretractionmethodfrom[82,Section1.2.4]

ePropBasic1.1rties

23

Proposition1.1.10.Letp∈(1,∞),µ∈(1/p,1],s≥0,α∈(0,2),andω>0.Thenthe
ratorseop(ω−∂t)αonHsp,µ(R+;E),withdomainHsp,µ+α(R+;E),
(ω−∂t)αonWsp,µ(R+;E),withdomainWsp,µ+α(R+;E),s,s+α∈/N0,
(ω+∂t)αon0Hsp,µ(R+;E),withdomain0Hsp,µ+α(R+;E),
(ω+∂t)αon0Wsp,µ(R+;E),withdomain0Wsp,µ+α(R+;E),s,s+α∈/N0,
areinvertibleandadmitaboundedH∞-calculuswithH∞-angleαπ/2,respectively.
Proof.(I)Wefirstconsiderthecases=0.Theorem1.1.7and[24,Proposition2.11]imply
thattherealizationofω−∂tonLp,µ(R+;E)withdomainW1p,µ(R+;E)admitsabounded
H∞-calculuswithH∞-angleequaltoπ/2.LemmaA.3.5yieldsthatalso(ω−∂t)αadmits
aboundedH∞-calculus,withH∞-angleαπ/2,providedα∈(0,2).Thesamearguments
asinStepIoftheproofofLemma1.1.9furthershowthat
D((ω−∂t)α)=Hαp,µ(R+;E).
(II)Sinceω−∂tisinvertible,also(ω−∂t)sisinvertible,foralls≥0.Itfollows
fromthedefinitionoftheweightedSobolevspacesthat(ω−∂t)sisanisomorphism
Hkp,µ+s(R+;E)→Hkp,µ(R+;E)fork∈N0,andbyinterpolationthiscarriesovertoan
isomorphismHτp,µ+s(R+;E)→Hτp,µ(R+;E)forallτ≥0.Since(ω−∂t)sand(ω−∂t)α
commute,itfollowsfrom[24,Proposition2.11]that(ω−∂t)αhasaboundedH∞-
calculusonHsp,µ(R+;E),stillwithanglenotlargerthanαπ/2,andthatitsdomainequals
Hsp,µ+α(R+;E).
(III)Nowlets,s+α∈/N0.ItthenfollowsfrominterpolationoftheH-caseandLemma
1.1.9that(ω−∂t)αhasaboundedH∞-calculusonWsp,µ(R+;E)withH∞-angleαπ/2,and
thatitsdomainisWsp,µ+α(R+;E).Thesameargumentsasaboveshowtheassertionsonthe
operatorω+∂t.
WeconsiderthetemporaltraceontheWsp,µ-spaces,andcharacterizationsofthe0Wsp,µ-
spacesintermsofitskernel.TheseresultsaremainlyduetoGrisvard[44].Observethatthe
limitnumberfortheexistenceofatraceiss=1−µ+1/p.Therefore,ifµrunsthroughthe
interval(1/p,1]thislimitnumberrunsthroughtheinterval[1/p,1).Ofcourse,forµ=1
thelimitnumbers=1/pfortheunweightedcaseisrecovered.
Proposition1.1.11.LetJ=(0,T)befiniteorinfinite,p∈(1,∞)andµ∈(1/p,1].
Thenthefollowingholdstrue.6
a)For0<s<1−µ+1/pitholdsCc∞(J\{0};E)→dWsp,µ(J;E),andfurther
Wsp,µ(J;E)=0Wsp,µ(J;E).
6Wedonottreatthelimitcasess=k+1−µ+1/p,k∈N0,sincetheyarequitecomplicatedandnot
importantforourpurposes.Forshortdiscussionswereferto[44,Remarque4.2]and[82,Remark3.6.3/2].
Wealsodonotconsiderthecorrespondingcharacterizationsofthe0Hsp,µ-spaces.Theyshouldbecorrect,
butitseemsthattheirproofsrequireagreatereffort.

24

TheSpacesLp,µandWeightedAnisotropicSpaces

b)Fork+1−µ+1/p<s<k+1+(1−µ+1/p)withk∈N0itholds
Wsp,µ(J;E)→BUCk(J;E),(1.1.13)
wherehereonemayreplaceWsp,µbyHsp,µ,andmoreover
0Wsp,µ(J;E)=∙u∈Wsp,µ(J;E):u(j)(0)=0,j∈{0,...,k},(1.1.14)
wherethelatterspaceisconsideredasaclosedsubspaceofWsp,µ(J;E).
Theembeddingconstantsfor
0Wsp,µ(J;E)→BUCk(J;E),0Hsp,µ(J;E)→BUCk(J;E)
wheres∈[0,2]andk∈N0areasinb),areindependentofJ,respectively.
Proof.Theresultsin[44]fortheWsp,µ(R+;E)-spacesareobtainedinthescalar-valued
case,E=C.Aninspectionoftheproofsthereshowsthat,besidesbasicfactsonvector-
valuedspaces,theyonlymakeuseofinterpolationtheoryandtheLemmas1.1.4and1.1.6.
Thustheresultsof[44]carryovertoageneralE.Moreover,thecaseofafiniteintervalis
obtainedfromthehalf-linecasebyextensionandrestriction,asinStepIVoftheproofof
Lemma1.1.9.ThefactthatonemayreplaceWbyHasassertedfollowsfrom(1.1.1)and
(1.1.2).Assertiona)isshownin[44,Théorème2.1,Théorème4.1].Theembeddinginb)isfork=0
provedin[44,Théorème5.2],andthegeneralcasek∈Nisanimmediateconsequence.For
s≤1,(1.1.14)isshownin[44,Théorème4.1].Fors>1,notethatbydefinitionitholds
0Wsp,µ(R+;E)→Wsp,µ(R+;E).
Fortheconverseembeddingin(1.1.14),takeu∈Wsp,µ(R+;E)withu(j)(0)=0forj∈
{0,...,k}.Assumefirstthat[s]=k.Thenu∈0W[p,µs](R+;E).Fromu([s])(0)=0,1−µ+
1/p<s−[s],(1.1.14)fors−[s]<1andwithLemma1.1.8weinfer
]s[|u|0Wsp,µ(R+;E)|u|0W[p,µs](R+;E)+|u|0Wsp,µ−[s](R+;E)
|u|W[p,µs](R+;E)+|u[s]|Wsp,µ−[s](R+;E)|u|Wsp,µ(R+;E).
]s[Nowassumethat[s]=k([+s])1.Thens−[s]againu∈0Wp,µ(sR−[+s];E).Sinces−[s]<1−µ+1/p,
itfollowsfroma)thatu∈Wp,µ(R+;E)=0Wp,µ(R+;E),and(1.1.14)followsas
abovefromLemma1.1.8.
WenextconsiderembeddingsofSobolevtypeintoweightedandunweightedspaces.
Proposition1.1.12.LetJ=(0,T)beafiniteinterval,p∈(1,∞),µ∈(1/p,1],s>τ≥0
andq∈(p,∞).Then
Wsp,µ(J;E)→Wqτ,µ(J;E)ifs−(1−µ+1/p)>τ−p(1−µ+1/p),(1.1.15)
qandmoreoveritholds
Wsp,µ(J;E)→Wqτ(J;E)ifs−(1−µ+1/p)>τ−1/q.(1.1.16)

ePropBasic1.1rties

25

TheseembeddingsremaintrueifonereplacestheW-spacesbytheH-,the0W-andthe
0H-spaces,respectively.Intwolattercases,restrictingtos∈[0,2],forgivenT0>0the
embeddingsholdwithauniformconstantforall0<T≤T0.
Proof.Throughoutthisproof,letT0>0begiven.Sincetheinequalitysignsin(1.1.15)
and(1.1.16)arestrict,wemayassumethats∈/N.Againweonlyhavetoconsiderthe
W-casedueto(1.1.1)and(1.1.2).
(I)Weshow(1.1.15)forτ=0.Fors>1−µ+1/p,theconditionissatisfiedforall
q∈(p,∞),and,infact,Proposition1.1.11shows
Wsp,µ(J;E)→L∞(J;E)→Lq,µ(J;E),q∈(p,∞),
withtheassertedbehaviouroftheembeddingconstantinthe0W-case.Fors≤1−µ+1/p
wetakeη>1−µ+1/panduseagainthatWηp,µ(J;E)→Lr,µ(J;E)forr∈(p,∞),A.2
d),Lemma1.1.9and[82,Theorem1.18.5],toobtain
Wsp,µ(J;E)=(Lp,µ(J;E),Wηp,µ(J;E))s/η,p→(Lp,µ(J;E),Lr,µ(J;E))s/η,p=Lq,µ(J;E),
whichisvalidforq1=1−ps/η+rs/η.Lettingr∞andη1−µ−1/p,weobtain
(1.1.15)forqasasserted.Inthe0W-case,theembeddingconstantisuniforminT≤T0
fors∈[0,2].
(II)Toprove(1.1.15)forτ>0,westartwith
Wsp,µ(J;E)=(Wκp,µ(J;E),Wηp,µ(J;E))ηs−−κκ,p,
whichholdsbyLemma1.1.9fornonintegerκ<s<η.Letk∈N0.Using(1.1.15)with
τ=0,weobtain
Wκp,µ(J;E)→Wqk,µ(J;E),κ>k+(1−p/q)(1−µ+1/p),
Wηp,µ(J;E)→Wqk,µ+1(J;E),η>(k+1)+(1−p/q)(1−µ+1/p).
Henceforthoseq,κ,ηitholds
κ−sWsp,µ(J;E)→(Wqk,µ(J;E),Wqk,µ+1(J;E))ηs−−κκ,q=Wqk,µ+η−κ(J;E),
usingthat(∙,∙)θ,p→(∙p,∙)θ,qforθ∈(0,1)andq∈(p,∞).Lettingκk+(1−qp)(1−µ+1/p)
andηk+1+(1−q)(1−µ+1/p)weobtain(1.1.15)forτandqasasserted.For0W-
spaces,thedependenceonTfors∈[0,2]carriesoverfromLemma1.1.9and(1.1.15)with
.0=τ(III)Toshow(1.1.16),weagainfirsttreatthecaseτ=0.Asabove,fors>1−µ+1/p
theembeddingisdeducedfromProposition1.1.11.Fors≤1−µ+1/pweuse
Hηp,µ(J;E)→L∞(J;E),η>1−µ+1/p,
thatfurtherandLp,µ(J;E)→Lr(J;E),1−µ+1/p<1/r,

26

TheSpacesLp,µandWeightedAnisotropicSpaces

whichfollowsfromLemma1.1.1.For0<σ<η,Lemma1.1.9andA.2d)andn)thusyield
Hσp,µ(J;E)→[Lr(J;E),L∞(J;E)]ησ=L1−σr/η(J;E).
Lettingη1−µ+1/p,r1−µ1+1/p,andemployingWsp,µ(J;E)→Hσp,µ(J;E)fors>σ
weobtain(1.1.16).ReplacingWby0W,theembeddingconstantisuniforminT.
(IV)Thecaseτ>0maynowbeobtainedfromthecaseτ=0asinStepII.Weomitthe
details.WederiveanintrinsicnormfortheW-spaces,onafiniteandaninfiniteinterval.
Proposition1.1.13.LetJ=(0,T)withT∈(0,∞],p∈(1,∞),µ∈(1/p,1],and
s∈(0,1).Thenwehave
|u|Wsp,µ(J;E)∼|u|Lp,µ(J;E)+[u]Wsp,µ(J;E),
whereTtp
[u]pWsp,µ(J;E):=τp(1−µ)|u(t)−u(1+τsp)|Edτdt.(1.1.17)
00(t−τ)
IncaseJ=R+,thesemi-norm[u]Wsp,µ(J;E)maybereplacedby
p∞∞[[u]]pWsp,µ(R+;E):=τp(1−µ)|u(t+τ)1+−spu(τ)|Edτdt.(1.1.18)
t00Proof.(I)ForJ=R+,itfollowsimmediatelyfromLemma1.1.6andA.2k)that
|u|Wsp,µ(J;E)∼|u|Lp,µ(J;E)+[[u]]Wsp,µ(J;E),
andasimplesubstitutionshowsthat[[u]]Wsp,µ(J;E)maybereplacedby[u]Wsp,µ(J;E).
(I)NowletJ=(0,T)befinite.Wededucethiscasefromthehalf-linecasebylocalization
andextension.Wefixasmoothpartitionofunity{ψ1,ψ2}for[0,T],suchthatψ1(t)=0
fort≥32Tandψ2(t)=0fort≤3T.Themultiplicationwithψi,i=1,2,iscontinuous
onLp,µ(J;E)andW1p,µ(J;E),respectively,henceitiscontinuousonWsp,µ(J;E)byA.2i).
impliesThis|u|Wsp,µ(J;E)|ψ1u|Wsp,µ(J;E)+|ψ2u|Wsp,µ(J;E).
(II)SincetherestrictiontoJiscontinuousonthewholeWsp,µ-scale,wemayestimate
|ψ1u|Wsp,µ(J;E)|ψ1u|Wsp,µ(R+;E)|ψ1u|Lp,µ(R+;E)+[ψ1u]Wsp,µ(R+;E),
identifyingψ1uwithitstrivialextensiontoR+.Wesplittheoutert-integralin
[ψ1u]pWsp,µ(R+;E)att=Tintotwosummands.Forthefirstsummandweestimate,using
themeanvaluetheoremforψ1,
tTτp(1−µ)|ψ1(t)u(t)−ψ1+1(spτ)u(τ)|pdτdt
00(t−τ)TT
[u]pWs(J;E)+τp(1−µ)|u(t)|p(t−τ)p(1−s)−1dtdτ
p,µτ0pp[u]Wsp,µ(J;E)+|u|Lp,µ(J;E).

rtiesePropBasic1.1

27

Forthesecondsummandwehave
∞tp(1−µ)|ψ1(t)u(t)−ψ1(τ)u(τ)|p
τ(t−τ)1+spdτdt
0T32T∞|ψ(τ)|p
=τp(1−µ)|u(τ)|p11+spdtdτ|u|Lpp,µ(J;E),
0T(t−τ)
sincetheintegralinbracketsisboundedindependentofτ∈(0,32T).
(III)ItfollowsfromA.2d)thatWps(J;E)→Wsp,µ(J;E),fromwhichweobtain
|ψ2u|Wsp,µ(J;E)|ψ2u|Wps(J;E)|ψ2u|Lp(J;E)+[ψ2u]Wps(J;E),
where[∙]Wps(J;E)denotestheintrinsicsemi-normintheunweightedcase(cf.(A.4.2)),i.e.,
pTT|ψ2(t)u(t)−ψ2(τ)u(τ)|Ep
[ψ2u]Wps(J;E)=00|t−τ|1+spdτdt.
Wesplittheinnerτ-integralof[ψ2u]pWs(J;E)atτ=tintotwosummands.Forthefirst
psummandwehave
tdτdTt|ψ2(t)u(t)−ψ2(τ)u(τ)|p
00(t−τ)1+sp
tTpτp(1−µ)|ψ2(τ)|p|u(t)−u(τ)|dτdt
T/3T/3(t−τ)1+sp
Ttp|ψ2(t)−ψ2(τ)|ppp
+T/3T/3|u(t)|(t−τ)1+spdτdt[u]Wsp,µ(J;E)+|u|Lp,µ(J;E).
Forthesecondsummandweestimateinasimilarfashion
TT|ψ2(t)u(t)−ψ2(τ)u(τ)|p
(τ−t)1+spdτdt
0tTτp
tp(1−µ)|ψ2(t)|p|u(τ)−u1+(tsp)|dtdτ
T/3T/3(τ−t)
Tτ|ψ2(t)−ψ2(τ)|ppp
+|u(τ)|p(τ−t)1+spdtdτ[u]Wsp,µ(J;E)+|u|Lp,µ(J;E).
T/3T/3
Theseestimatesshow|u|Wsp,µ(J;E)|u|Lp,µ(J;E)+[u]Wsp,µ(J;E).
(IV)Fortheconverseestimate,notethatittriviallyholds
pp∞t|EJu(t)−EJu(x)|pp
[u]Wsp,µ(J;E)≤[u]Wsp,µ(J;E)+τp(1−µ)(t−τ)1+spdτdt=[EJu]Wsp,µ(R+;E),
0TwhereEJistheextensionoperatorfromLemma1.1.5.Wethusobtain
|u|Lp,µ(J;E)+[u]Wsp,µ(J;E)≤|EJu|Wsp,µ(R+;E)|u|Wsp,µ(J;E),
whichfinishestheproof.
WenextprovePoincaré’sinequalityintheweightedspaces.Itwillbeusedinlaterchapters
toobtainsmallnessofLipschitzconstantsbychoosingshorttimeintervals.

28

TheSpacesLp,µandWeightedAnisotropicSpaces

Lemma1.1.14.LetJ=(0,T)befinite,p∈(1,∞),andµ∈(1/p,1].Thenitholds
|u|Lp,µ(J;E)T|u|Lp,µ(J;E)ifu∈0W1p,µ(J;E),
andconsequently,fors∈[0,1),
|u|0Wsp,µ(J;E)+|u|0Hsp,µ(J;E)T1−s|u|W1p,µ(J;E)ifu∈0W1p,µ(J;E).
Proof.Fort∈Jweestimate,usingHölder’sinequality,
pTtp(1−µ)|u(t)|Ep≤tp(1−µ)s−(1−µ)s1−µ|u(s)|Eds
0tp(1−µ)T(1−p(1−µ))p/p|u|Lp(J;E).
p,µNowthefirstassertedinequalityfollowsafterintegrationoverJ.Fors∈[0,1)theinter-
polationinequalityA.2j)yields
1s−1s|u|0Wsp,µ(J;E)+|u|0Hsp,µ(J;E)|u|0W1p,µ(J;E)|u|Lp,µ(J;E),u∈0Wp,µ(J;E),
fromwhichthesecondassertedinequalityfollows.

Remark1.1.15.Inapplicationsonedealswithsuperpositionandmultiplicationoperators
onthespacesWsp,µ(0,T;E)and0Wsp,µ(0,T;E),equippedwiththeinterpolationnormfrom
theirdefinitioninthebeginningofthissection.Ofcourse,onewouldratherliketowork
withtheintrinsicnormsderivedinProposition1.1.13,sincethesemuchmoreconvenient
toworkwith.AtthesametimeoneoftenassumesthatTissmall,forinstancetomake
lowerordertermssmall,withPoincaré’sinequality(seeLemma1.3.13).Suchascenario
arises,forinstance,intheproofsofourmainTheorems2.1.4and3.1.4onlinearproblems,
andalsointheproofofProposition4.3.2onlocalexistencefornonlinearproblems.
InProposition1.1.13wehaveshowntheequivalenceoftheinterpolationnormandthe
intrinsicnormforWsp,µ(0,T;E)usingtheextensionoperatorEJfromLemma1.1.5.Thus
theequivalenceconstantsforthesenormsdependonT,andtypicallybecomelargeasT
becomessmall.Thismighthavetheeffectthatlowerorderterms,forinstance,arenot
smallanymoreforsmallTafterhavingusedtheintrinsicnorm.Thesituationisthesame
ifoneworksin0Wsp,µ(0,T;E)equippedwiththeintrinsicnorm.
Toovercomethisobstacleforshorttimeintervals,inasituationasaboveonehastoworkin
0Wsp,µ(0,T;E)equippedwiththeinterpolationnormfromthebeginning.Viatheextension
operatorEJ0andrestriction,thisspaceisT-independentlyconnectedto0Wsp,µ(R+;E).In
thiswayonecanperformtherequiredestimateswiththeintrinsicnorms(1.1.17)or(1.1.18)
on0Wsp,µ(R+;E),withoutreceivingunpleasantT-dependentfactors.Forexampleswerefer
totheproofsoftheLemmas1.3.22,1.3.23and4.2.3,forinstance.

ertiesPropAbstract1.21.2.1AbstractMaximalLp,µ-Regularity
WebrieflyreviewtheresultsofPrüss&Simonett[71]onabstractmaximalLp,µ-regularity,
andaddafewremarksonfiniteintervals.

ertiesPropAbstract1.2

29

LetAbeaclosedanddenselydefinedoperatoronaBanachspaceEwithdomainD(A).
Endowedwithitsgraphnorm,D(A)becomesaBanachspace.Letfurtherp∈(1,∞)and
µ∈(1/p,1].ForafiniteorinfiniteintervalJ=(0,T)weset
E0,µ(J):=Lp,µ(J;E),Eu,µ(J):=W1p,µ(J;E)∩Lp,µ(J;D(A)).
DuetoLemma1.1.1,functionsinEu,µ(J)haveawell-definedtraceinEatt=0.Wesay
thatAenjoysmaximalLp,µ-regularityonJ,
A∈MRp,µ(J;E),
ifforeachf∈E0,µ(J)thereisauniquesolutionu∈Eu,µ(J)of
u+Au=f(t),a.e.t∈J,u(0)=0.(1.2.1)
Inotherwords,itholdsA∈MRp,µ(J;E)ifandonlyiftheoperator∂t+AonE0,µ(J),
domainwith0Eu,µ(J):=0W1p,µ(J;E)∩Lp,µ(J;D(A)),
isinvertible.Forconveniencewefurtherset
MRp(J;E):=MRp,1(J;E)
intheunweightedcase.IfA∈MRp,µ(J;E),thentheopenmappingtheoremimpliesthat
thesolutionuof(1.2.1)dependscontinuouslyontheright-handsidef,i.e.,thereisa
constantC>0,whichdoesnotdependonf,suchthat
|u|Eu,µ(J)≤C|f|E0,µ(J).(1.2.2)
Thefollowinglemmashowsthatfornegativegeneratorsofanalyticsemigroups,maximal
Lp,µ-regularityisonlyamatterofregularity,sincethesolutionof(1.2.1)isgivenbythe
convolutionwiththesemigroup.
Lemma1.2.1.LetJ=(0,T)befiniteorinfinite,p∈(1,∞),µ∈(1/p,1],andlet−Abe
thegeneratorofananalyticsemigrouponE.Ifu∈Eu,µ(J)solves(1.2.1)forf∈E0,µ(J),
thenuisgivenby
u(t)=te−(t−s)Af(s)ds,t∈J.
0Inparticular,Eu,µ(J)-solutionsof(1.2.1)areunique.
Proof.ByLemma1.1.1itholdsLp,µ(J;E)→L1,loc(J;E),andthustheassertionfollows
immediatelyfrom[30,Theorem2.1].
Thefollowingfundamentalresultdueto[71]showsthatthemaximalregularityproperties
ofAonthehalf-lineareindependentoftheweight.
Theorem1.2.2.Forp∈(1,∞)andµ∈(1/p,1]itholdsA∈MRp,µ(R+;E)ifandonly
ifA∈MRp(R+;E).Moreover,ifA∈MRp(J0;E)forsomefiniteorinfiniteinterval
J0=(0,T0)thenA∈MRp,µ(J;E)forallµ∈(1/p,1]andallfiniteintervalsJ=(0,T)
aswell,andifA∈MRp(R+;E)thentheconstantin(1.2.2)isindependentofJ.

30

TheSpacesLp,µandWeightedAnisotropicSpaces

Proof.(I)TheindependenceoftheclassMRp,µ(R+;E)ofµ∈(1/p,1]isshownin[71,
2.4].Theorem(II)AssumethatA∈MRp(J0;E).Itthenfollowsfrom[30,Corollary5.3]thatthere
isω>0suchthatA−ω∈MRp(R+;E),andthusA−ω∈MRp,µ(R+;E)forall
µ∈(1/p,1]bytheresultof[71].Itcannowbeshownasintheproofof[30,Theorem3.3]
thatA∈MRp,µ(J;E)foreachfiniteintervalJ=(0,T).
(III)Finally,supposethatA∈MRp(R+;E),andletJbefinite.Forf∈Lp,µ(J;E)a
solutionu∈Eu,µ(J)of
u+Au=f(t),a.e.t∈J,u(0)=0,
isgivenbyu=u|J,whereuisthesolutionoftheaboveproblemonR+withtrivially
extendedright-handsidef.Since−AisthegeneratorofananalyticC0-semigrouponEby
[30,Corollary4.2],itfollowsfromLemma1.2.1thatthisistheonlysolution.Thisyields
estimatethe|u|Eu,µ(J)≤|u|Eu,µ(R+)≤C|f|E0,µ(R+)=C|f|E0,µ(J),
whereCisthemaximalregularityconstantofAonR+,whichisindependentofJ.
WedescribesomeconsequencesofTheorem1.2.2formaximalLp,µ-regularity.
IfEisofclassHT,thenwellknownsufficientconditionsformaximalLp-regularityare
alsoavailableformaximalLp,µ-regularity,suchasthatAadmitsaboundedH∞-calculus
oradmitsboundedimaginarypowers,withanglesstrictlysmallerthanπ/2,respectively.
Moreover,combiningTheorem1.2.2witharesultofWeis[85,Theorem4.2],itholds
A∈MRp,µ(R+;E)ifandonlyifAisR-sectorial.
From[30,Theorem7.1]itfollowsthatmaximalLp,µ-regularityisindependentofthe
exponentp∈(1,∞),andtogetherwith[30,Corollary4.2]wefurtherobtainthatif
A∈MRp,µ(R+;E),then−AisthegeneratorofanexponentiallystableanalyticC0-
.EonsemigroupNowletusconsider(1.2.1)withnontrivialinitialvalues,i.e.,
u+Au=f(t),a.e.t∈J,u(0)=u0.(1.2.3)
Thefollowingresultisprovedin[71,Theorem3.2]forJ=R+.Thecaseoffiniteinterval
maybededucedfromthisasintheproofofTheorem1.2.2.
Theorem1.2.3.Letp∈(1,∞),µ∈(1/p,1],andletJ=(0,T)befiniteorinfinite.If
A∈MRp(R+;E)then(1.2.3)hasauniquesolutionu∈Eu,µ(J)ifandonlyiff∈E0,µ(J)
andu0∈DA(µ−1/p,p).7ThereisaconstantC,whichisindependentofJ,f,andu0,
thathsuc|u|Eu,µ(J)≤C|f|E0,µ(J)+|u0|DA(µ−1/p,p).
7RecallthenotationDA(µ−1/p,p)=(E,D(A))µ−1/p,p.

1.2ertiesPropAbstract

31

1.2.2Operator-ValuedFourierMultipliers
Wenowturnourattentiontooperator-valuedFouriermultipliersonLp,µ.ForBanach
spacesE,Fandanoperator-valuedfunctionm∈L1,loc(R;B(E,F))oneobtainsanoper-
atorTmbysetting
Tmf:=F−1mFf,f∈F−1Cc∞(R;E),
whereFdenotestheFouriertransformonR.ItisnothardtoshowthatTmisdensely
definedonLp,µ(R+;E).NowmiscalledaFourier-multiplieronLp,µ,iftheoperatorTm
estimateanadmits|Tmf|Lp,µ(R+;F)|f|Lp,µ(R+;E),f∈F−1Cc∞(R;E),
i.e.,ifitextendstoacontinuousoperatorfromLp,µ(R+;E)toLp,µ(R+;F).
ThefollowingresultonLp,µ-multipliersisavailable.ItisduetoGirardiandWeis[42],and
isanextensionofWeis’multipliertheorem[85,Theorem3.4]intheunweightedcase.
Theorem1.2.4.Letp∈(1,∞),µ∈(1/p,1],andletEandFbeBanachspacesofclass
HT.Assumethatm∈C1R\{0};B(E,F)satisfies
R{m(λ),λm(λ):λ=0}≤κ.
ThenTm∈BLp,µ(R+;E),Lp,µ(R+;F),withnormnotexceedingC(p,µ,X,Y)κ.
Weremarkthatacorrespondingtheoremholdstrueinarbitrarydimensions,andformore
generalweightsfromtheclassAp.

UndermorerestrictiveassumptionsonmwecangiveashortproofofTheorem1.2.4,using
aresultofKreé[60]whichisalsothebasisforthetheoremof[42].
Proposition1.2.5.UndertheassumptionsofTheorem1.2.4,letmsatisfyinaddition
m∈C2R\{0};B(E,F),suchthat
|m(λ)|B(E,F)|λ|−2,λ=0.
ThenTmextendstoacontinuousoperatorfromLp,µ(R+;E)toLp,µ(R+;F).
Proof.Itfollowsfromtheoperator-valuedmultipliertheoremintheunweightedcasethat
TmextendstoaboundedoperatorfromLp(R+;E)toLp(R+;F).
Moreover,followingthelinesoftheproofof[81,LemmaVI.4.4.2],theassumptionson
myieldthatTmmayberepresentedasaconvolutionoperator,withakernelk∈
CR\{0};B(E,F),satisfying|k(t)|B(E,F)|t1|.
Itnowfollowsfrom[60,Théorème2]thatTmisalsoboundedfromLp,µ(R+;E)to
Lp,µ(R+;F),forallµ∈(1/p,1].

32

TheSpacesLp,µandWeightedAnisotropicSpaces

1.3WeightedAnisotropicSpaces
LetEbeaBanachspaceofclassHT,letJ=(0,T)befiniteorinfinite,andletfurther
Ω⊂Rnbeadomainwithcompactsmoothboundary∂Ω,orΩ∈{Rn,R+n}.Inwhatfollows
werefertot∈Jastimevariables,andtox∈Ωasspacevariables.Forp,q∈[1,∞]and
r>0wedenoteby
Hpr(Ω;E),Wpr(Ω;E),Brp,q(Ω;E),
theE-valuedBesselpotential,SlobodetskiiandBesovspaces.RecallthatBrp,p(Ω;E)=
Wpr(Ω;E)forp∈[1,∞)andr∈/N0.Thecorrespondingspacesovertheboundary∂Ωare
definedvialocalcharts.WerefertoAppendixA.4fordefinitionsandpropertiesofthese
spaces.function

Inthissectionweinvestigateweightedanisotropicspaces,i.e.,intersectionsofspacesof
formtheHsp,µJ;Hpr(Ω;E),Wsp,µJ;Wpr(Ω;E),Hsp,µJ;Wpr(Ω;E),Wsp,µJ;Wpr(Ω;E),

(1.3.1)wheres,r≥0.WearefurtherconcernedwiththecorrespondingspacesoverJ×∂Ω,
andwithintersectionsofspaceswherein(1.3.1)Hsp,µandWsp,µarereplacedby0Hsp,µand
0Wsp,µ,respectively.WeconsidertheNewtonpolygon,temporalandspatialtracetheorems,
andsufficientconditionsforpointwisemultipliersforthesespaces.

Westartwithtwofundamentaltoolsforanisotropicspaces.

Thefirstisaspatialextensionoperator.Givenk∈N,thereisanextensionoperatorEΩ
toRnforfunctionsdefinedonΩ,i.e.,(EΩu)|Ω=u,suchthatforallp,q∈(1,∞)and
r∈[0,k]itholds
EΩ∈BBrp,q(Ω;E),Brp,q(Rn;E)∩BHpr(Ω;E),Hpr(Rn;E).(1.3.2)
Forintegerr∈[0,k],theproofof[1,Theorems5.21,5.22]forthescalar-valuedspaces
literallycarriesovertothevector-valuedcase.Thegeneralcaser∈[0,k]followsfrom
interpolation.ApplyingEΩpointwisealmosteverywhereintime,weobtainaspatialex-
tensionoperatorfortheanisotropicspaces,whichwedenotebyEΩagain,
EΩ∈BHsp,µ(J;Hpr(Ω;E)),Hsp,µ(J;Hpr(Rn;E)),s≥0,r∈[0,k].(1.3.3)
Ofcourse,hereaH-spacemaybereplacedbyaW-spaceatthefirstorthesecondorat
bothpositions,andthisremainstrueforthe0Hsp,µ-andthe0Wsp,µ-spaceswithrespectto
time.

Second,weconsideroperatorswithboundedimaginarypowers(cf.AppendixA.3)onthe
weightedanisotropicspacesforthecaseJ×Ω=R+×Rn.Thisclassofoperatorsiscrucial
forourpurposes,inviewoftheDore-VenniTheoremA.3.2andYagi’stheorem(A.3.1).

eightedW1.3SpacesAnisotropic

33

Lemma1.3.1.LetEbeofclassHT,p∈(1,∞),µ∈(1/p,1],s,r≥0,α∈(0,2)and
β>0,letω,ω≥0satisfyω+ω=0andset
Hsp,µ(Hpr):=Hsp,µR+;Hpr(Rn;E),0Hsp,µ(Hpr):=0Hsp,µR+;Hpr(Rn;E),
andanalogouslyfortheothertypesofspacesin(1.3.1).LetΔnbetheLaplacianonRn.
Thenthefollowingholdstrue.
a)Thepointwiserealizationof(ω−Δn)β/2onthespaces
Hsp,µ(Hpr),withdomainHsp,µ(Hpr+β),
Hsp,µ(Wpr),withdomainHsp,µ(Wpr+β),r,r+β∈/N0,
Wsp,µ(Hpr),withdomainWsp,µ(Hpr+β),
Wsp,µ(Wpr),withdomainWsp,µ(Wpr+β),r,r+β∈/N0,
isinvertibleandadmitsaboundedH∞-calculuswithH∞-angleequaltozero.This
remainstrueifonereplacestheHsp,µ,Wsp,µ-spacesbythe0Hsp,µ,0Wsp,µ-spaces.
b)OnLp,µ(Lp),theoperators(ω−∂t)αand(ω+∂t)αcommutewith(ω−Δn)β/2in
theresolventsense,respectively.
c)TheoperatorL:=(ω−∂t)α+(ω−Δn)β/2,consideredonthespaces
Hsp,µ(Hpr),withdomainHsp,µ+α(Hpβ)∩Hsp,µ(Hpr+β),
Hsp,µ(Wpr),withdomainHsp,µ+α(Wpβ)∩Hsp,µ(Wpr+β),r,r+β∈/N0,
Wsp,µ(Hpr),withdomainWsp,µ+α(Hpβ)∩Wsp,µ(Hpr+β),s,s+α∈/N0,
Wsp,µ(Wpr),withdomainWsp,µ+α(Wpβ)∩Wsp,µ(Wpr+β),s,s+α,r,r+β∈/N0,
isinvertibleandadmitsboundedimaginarypowers,withpoweranglenotlargerthan
απ/2,respectively.ThisremainstruefortheoperatorL0:=(ω+∂t)α+(ω−Δn)β/2
ifonereplacestheHsp,µ,Wsp,µ-spacesbythe0Hsp,µ,0Wsp,µ-spaces.
d)Forτ∈(0,1]itholds
D(Lτ)=D((ω−∂t)ατ)∩D((ω−Δn)βτ/2),
DL(τ,p)=D(ω−∂t)α(τ,p)∩D(ω−Δn)β/2(τ,p),
andthisremainstrueifonereplacesLbyL0andω−∂tbyω+∂t.
Proof.(I)SinceEisofclassHT,theoperator−ΔnadmitsonLp(Rn;E)withdomain
Hp2(Rn;E)aboundedH∞-calculuswithH∞-angleequaltozero,dueto[24,Theorem5.5],
forinstance.Thisremainsvalidfor(ω−Δn)β/2withdomainHpβ(Rn;E),duetoLemma
A.3.5and(A.3.1),andfurtherthisoperatorisinvertible.
Using(ω−Δn)r/2asanisomorphismbetweenHpr(Rn;E)andLp(Rn;E),itfollowsfrom
[24,Proposition2.11]that(ω−Δn)β/2hasthesamepropertiesonHpr(Rn;E),withdomain
Hpr+β(Rn;E),r≥0.Byinterpolation,thesefactsremaintrueifoneconsiders(ω−Δn)β/2

34

TheSpacesLp,µandWeightedAnisotropicSpaces

onWpr(Rn;E),withdomainWpr+β(Rn;E),providedr,r+β∈/N0.DuetoLemmaA.3.6,
thesepropertiescarryovertoitspointwiserealizations.
(II)Theexplicitrepresentationoftheresolventsofω−∂tandω+∂t(see,forinstance,[48,
Proposition8.4.1])yieldsthatonLp,µ(Lp)theseoperatorsareresolventcommutingwith
ω−Δn,respectively.By[7,LemmaIII.4.9.2],thispropertycarriesovertothefractional
case.erwop(III)SinceallthespacesunderconsiderationareofclassHT,itfollowsfromProposition
1.1.10that(ω−∂t)αadmitsaboundedH∞-calculuswithH∞-angleequaltoαπ/2on
Hsp,µ(Hpr)withdomainHsp,µ+α(Hpr),andonthecorrespondingspaceswhereHisreplaced
byW,withtheassertedexceptions.Usingthisfact,togetherwitha)andb),theassertions
onLareaconsequenceoftheDore-VenniTheoremA.3.2.Thesameargumentsshowthe
assertiononL0.Finally,d)isaconsequenceoftheLemmasA.3.1andA.3.4.

olygonPNewtonThe1.3.1WiththehelpoftheoperatorsfromLemma1.3.1weestablishfundamentalembeddings
fortheanisotropicspaces.Thecorrespondingresultsforexponentiallyweightedspacesare
obtainedin[27,Lemma4.3].
Proposition1.3.2.LetEbeofclassHT,letJ=(0,T)befiniteorinfinite,p∈(1,∞),
µ∈(1/p,1],andletΩ⊂Rnbeadomainwithcompactsmoothboundary∂Ω,orΩ∈
{Rn,R+n}.Letfurther
s,r≥0,α∈(0,2),β>0,σ∈[0,1],
andsetHsp,µ(Hpr):=Hsp,µJ;Hpr(Ω;E),andanalogouslyfortheotheranisotropicspaces.
holdsitThenHsp,µ+α(Hpr)∩Hsp,µ(Hpr+β)→Hsp,µ+σα(Hpr+(1−σ)β),(1.3.4)
andmoreovereachofthespaces
Hsp,µ+α(Wpr)∩Hsp,µ(Wpr+β),Wsp,µ+α(Hpr)∩Wsp,µ(Hpr+β),Wsp,µ+α(Hpr)∩Hsp,µ(Wpr+β),
iscontinuouslyembeddedin
Wsp,µ+σα(Hpr+(1−σ)β)∩Hsp,µ+σα(Wpr+(1−σ)β),
providedalltheoccurringWp,µ-andW-spaceshaveanonintegerorderofdifferentiability.
Finally,assumingallordersofdifferentiabilitytobenoninteger,itholds
Wsp,µ+α(Wpr)∩Wsp,µ(Wpr+β)→Wsp,µ+σα(Wpr+(1−σ)β).(1.3.5)
TheseembeddingsremaintrueifonereplacesΩbyitsboundary∂Ω.Theyalsoremain
trueifonereplacesalltheHp,µ-,Wp,µ-spacesbythe0Hp,µ-,0Wp,µ-spaces.Restrictingin
thelattercasetos+α≤2,theembeddingconstantsdonotdependonthelengthofJ.

SpacesAnisotropiceightedW1.3

35

Proof.(I)Usingextensionsandrestrictions,andemployingthatthespacesover∂Ωare
definedvialocalcharts,itsufficestoconsiderthecaseJ×Ω=R+×Rn.Thedependenceof
theembeddingconstantsonJcarriesoverfromthepropertiesoftheextensionoperators.
(II)For(1.3.4)weconsidertheoperators(1−∂t)αand(1−Δn)β/2onHsp,µ(Hpr),which
weretreatedinProposition1.1.10andLemma1.3.1.Notethattoobtainsectorialityof
(1−∂t)αwehavetorestricttoα∈(0,2).Duetotheinvertibilityoftheseoperators,for
σ∈(0,1)itholdsthat
|(1−∂t)ασ(1−Δn)β(1−σ)/2∙|Hsp,µ(Hpr)
isanequivalentnormonHsp,µ+σα(Hpr+(1−σ)β).Sincethesumoftheseoperatorsisinvertible
byLemma1.3.1,itfurtherholdsthat
|((1−∂t)α+(1−Δn)β(1−σ)/2)∙|Hsp,µ(Hpr)
isanequivalentnormonHsp,µ+α(Hpr)∩Hsp,µ(Hpr+β).Now(1.3.4)followsfromthemixed
derivativetheorem,LemmaA.3.3.Thesameargumentsshow
Hsp,µ+α(Wpr)∩Hsp,µ(Wpr+β)→Hsp,µ+σα(Wpr+(1−σ)β),
Wsp,µ+α(Hpr)∩Wsp,µ(Hpr+β)→Wsp,µ+σα(Hpr+(1−σ)β),
and(1.3.5),withtheindicatedexceptions.Inthefollowingwederivetheremainingem-
beddingsfrom(1.3.4)bysuitableinterpolationarguments,whichwereindicatedin[37,
Remark5.3]inamorespecialsituation.
(III)ForHsp,µ+α(Wpr)∩Hsp,µ(Wpr+β)wesupposethatr,r+β∈/N0.Weapplythereal
interpolationfunctor(∙,∙)1/2,ptotheembedding
Hsp,µ+α(Hpr±ε)∩Hsp,µ(Hpr±ε+β)→Hsp,µ+α(σ±ε/β)(Hpr+(1−σ)β),(1.3.6)
whereε>0issufficientlysmall.ByLemma1.1.9theright-handsidesinterpolateto
Wsp,µ+σα(Hpr+(1−σ)β).Tointerpolatetheleft-handsidesaboveweconsidertheoperator
L=(1−∂t)α+(1−Δn)β/2,
which,duetoLemma1.3.1,isanisomorphism
Hsp,µ+α(Hpr±ε)∩Hsp,µ(Hpr±ε+β)→Hsp,µ(Hpr±ε).
HenceLisanisomorphismbetween
Hsp,µ+α(Hpr−ε)∩Hsp,µ(Hpr−ε+β),Hsp,µ+α(Hpr+ε)∩Hsp,µ(Hpr+ε+β)1,p
2andHsp,µ(Hpr−ε),Hsp,µ(Hpr+ε)21,p,andthelatterspaceequalsHsp,µ(Wpr),duetoLemma
1.1.9andPropositionA.4.2.ByLemma1.3.1,theoperatorL−1mapsHsp,µ(Wpr)isomor-
tophicallyHsp,µ+α(Wpr)∩Hsp,µ(Wpr+β).

36

TheSpacesLp,µandWeightedAnisotropicSpaces

Thuswehaveshownthatthelefthandsidein(1.3.6)interpolatestoHsp,µ+α(Wpr)∩
Hsp,µ(Wpr+β).ForWsp,µ+α(Hpr)∩Wsp,µ(Hpr+β)wehaves,s+α∈/N0.Hereweapply(∙,∙)1/2,p
toHsp,µ±ε+α(Hpr)∩Hsp,µ±ε(Hpr+β)→Hsp,µ+σα(Hpr+β(1−σ±ε/α)).
UsingtheoperatorLasaboveyieldstheassertedembeddinginthiscase.
(IV)ForWsp,µ+α(Hpr)∩Hsp,µ(Wpr+β)wehaves+α,r+β∈/N.Thistimeweapply(∙,∙)1/2,p
eddingsbemthetoHsp,µ+α(1±ε/β)(Hpr)∩Hsp,µ(Hpr+β±ε)→Hsp,µ+σα(Hpr+(1−σ)β±ε),
Hsp,µ+α(1±ε/β)(Hpr)∩Hsp,µ(Hpr+β±ε)→Hsp,µ+α(σ±ε/β)(Hrp+(1−σ)β).
Asaboveitfollowsthattheright-handsidesinterpolatetoHsp,µ+σα(Wpr+(1−σ)β)and
Wsp,µ+σα(Hpr+(1−σ)β),respectively.Tointerpolatetheleft-handside,weconsideronHsp,µ(Hpr)
eratoroptheL=(1−∂t)α(1+ε/β)+(1−Δ)(β+ε)/2,
withdomainD(L)=Hsp,µ+α(1+ε/β)(Hpr)∩Hsp,µ(Hpr+β+ε).DuetotheLemmas1.1.9,1.3.1
andPropositionA.4.2itholds
D(L(β−ε)/(β+ε))=Hsp,µ+α(1−ε/β)(Hpr)∩Hsp,µ(Hpr+β−ε),
yieldstheoremreiterationtheand(D(L(β−ε)/(β+ε)),D(L))1/2,p=DL((1+(β−ε)/(β+ε))/2,p).
sFinally+α,rthesLemmasr+β1.1.9,1.3.1andPropositionA.4.2implythatthelatterspaceequals
Wp,µ(Hp)∩Hp,µ(Wp).
(V)StartinginStepIIwith(1+∂t)αinsteadof(1−∂t)α,thesameargumentsasabove
showthattheassertedembeddingsarealsotrueforthe0Hp,µ-and0Wp,µ-spaces.

Remark1.3.3.Theproofshowsthatfortheembeddingswhereonlythemixedderiva-
tivetheoremwasusedtheordersofintegrabilityinspaceandtimedonothavetocoin-
cide.Infact,consideringtheLaplacianonHqr(Rn;E)forq∈(1,∞),andrealizingiton
Hsp,µ(J;Hqr(Rn;E))forp∈(1,∞)andµ∈(1/p,1],theassertionsofLemma1.3.1remain
true.ThenasinStepIIoftheaboveproofweobtain,forinstance,
Hsp,µ+α(Hqr)∩Hsp,µ(Hqr+β)→Hsp,µ+σα(Hqr+(1−σ)β),
Wsp,µ+α(Wqr)∩Wsp,µ(Wqr+β)→Wsp,µ+σα(Wqr+(1−σ)β),
withnonintegerordersofdifferentiabilityintheW-caseanduniformembeddingsinthe
0Hp,µ-and0Wp,µ-case.

eightedW1.3SpacesAnisotropic

37

Theaboveembeddingsturnouttobeextremelyusefulinthesequel.Theycanbevisualized
bytheNewtonpolygon.SupposethatananisotropicspaceXoftheform
mX=HpsjJ;Hprj(Ω;E),
=1jwhere0≤r1<...<rmandsj≥0,isgiven.ConsidereachspaceHpsj(J;Hprj(Ω;E))asa
point(rj,sj)inaspace-time-regularitydiagram,anddrawtheconvexhullofthesepoints
withrespecttotheboundaryofthepositivecone.
snontrivialpart(bold)

rFigure1.1:TheNewtonpolygon

ThishulliscalledtheNewtonpolygonNPforX,andthelinesonthehullconnectingpoints
(rj,sj)(includingthesepoints)iscalledthenontrivialpartofNP.Proposition1.3.2and
trivialembeddingsinspaceandtimeyieldthatXembedsintoeachspaceHps(J;Hpr(Ω;E))
forwhich(r,s)liesinsidetheNewtonpolygon.Ofcourse,hereonemayreplacetheH-
spacesbytheW-spacesaccordingtotheaboveresult.
AtypicalapplicationofProposition1.3.2isthefollowingproofofthemappingbehaviour
ofthespatialderivativeonanisotropicH-spaces.See[24,Lemma3.8]fortheunweighted
case.Lemma1.3.4.LetEbeaBanachspaceofclassHT,letJ=(0,T)befiniteorfinite,and
letΩ⊂Rnbeadomainwithcompactsmoothboundary,orΩ∈{R+n,Rn}.Letfurther
s≥0,r∈[0,1),α∈(0,2),β≥1.
Thenthepointwiserealizationof∂xi,i∈{1,...,n},isacontinuousmap
Hsp,µ+α(Hpr)∩Hsp,µ(Hpr+β)→Hsp,µ+α−α/β(Hpr)∩Hsp,µ(Hpr+β−1).
Restrictingtos+α≤2,andfurtherto0Hsp,µ+α-and0Hsp,µ-spacesintime,itsoperator
normisindependentofthelengthofJ.
Proof.ByextensionandrestrictionitsufficestoconsiderthecaseJ×Ω=R+×Rn.
Clearlytheoperator∂ximapscontinuously
Hsp,µ+α(Hpr)∩Hsp,µ(Hrp+β)→Hsp,µ(Hpr+β−1).
ItfurtherfollowsfromProposition1.3.2thattheembedding
Hsp,µ+α(Hpr)∩Hsp,µ(Hrp+β)→Hsp,µ+α−α/β(Hpr+1)

38

TheSpacesLp,µandWeightedAnisotropicSpaces

isvalid,andthus∂xialsomaps
Hsp,µ+α(Hpr)∩Hsp,µ(Hpr+β)→Hsp,µ+α−α/β(Hpr)
inacontinuousway.

1.3.2TemporalTraces
Wenowconsiderthetemporaltraceforanisotropicspaces.Usingintegrationbyparts,it
isnothardtoseethatforaBanachspaceXandu∈W11,loc([0,∞);X)therepresentation
σσt
u(0)=(2−µ)σ−(2−µ)τ1−µu(τ)dτ−(2−µ)t−(3−µ)τ1−µ(u(t)−u(τ))dτdt
000(1.3.7)holdstrueforallσ>0.ByA.2l),if−Aisthegeneratorofanexponentiallystableanalytic
C0-semigroupthenforθ∈(0,1)thenorminDA(θ,p)isequivalentto|∙|DA(θ,p),∗,where
∞σd|x|pDA(θ,p),∗=σp(1−θ)|Ae−σAx|pXσ.(1.3.8)
0Therepresentation(1.3.7)isthekeytothefollowingabstracttracetheorem,whoseproof
[29].BlasioDiwsfolloLemma1.3.5.LetXbeaBanachspace,p∈(1,∞),µ∈(1/p,1],andlettheoperatorA
onXwithdomainD(A)beinvertibleandadmitboundedimaginarypowerswithpower
anglestrictlysmallerthanπ/2.Lets∈(0,1−µ+1/p)andα>0satisfys+α∈
(1−µ+1/p,1).Thenthetemporaltracetr0,i.e.,tr0u=u(0),mapscontinuously
Wsp,µ+αR+;D(As)∩Wsp,µR+;D(As+α)→DA2s+α−(1−µ+1/p),p.(1.3.9)

Moreover,tr0isforα∈(1−µ+1/p,1]continuous
Wαp,µR+;X∩Lp,µR+;DA(α,p)→DAα−(1−µ+1/p),p,(1.3.10)
andfors∈(0,1−µ+1/p)itiscontinuous
W1p,µR+;DA(s,p)∩Wsp,µR+;D(A)→DA1+s−(1−µ+1/p),p.8(1.3.11)

Proof.Theproofsof(1.3.10)and(1.3.11)areverysimilartotheLemmas11and12of
[29],startingwith(1.3.7)andusingtherepresentation(1.1.17)oftheweightedSlobodetskii
seminormandHardy’sinequality(Lemma1.1.2).Wethereforeconcentrateon(1.3.9).
ByassumptionandProposition1.1.11itholds
|u(0)|X|u|Wsp,µ+α(R+;D(As)),u∈Wsp,µ+α(R+;D(As)).
8Theproofsof(1.3.10)and(1.3.11)onlyrequirethat−Ageneratesanexponentiallystableanalytic
-semigroup.C0

SpacesAnisotropiceightedW1.3

39

Wefurtheruse(1.3.7)and(1.3.8)toobtain
∞pp|u(0)|DA(2s+α−(1−µ+1/p),p),∗=σp(1−(2s+α−(1−µ+1/p))|Ae−σAu(0)|Xσ−1dσ
0∞σp
σ−p(2s+α)τ1−µ|Ae−σAu(τ)|dτdσ(1.3.12)
0∞0σtp
+σ2−µ−(2s+α)t−(3−µ)τ1−µ|Ae−σA(u(t)−u(τ))|dτdtdσ.
000ItfollowsfromA.2i)thatforθ∈(0,1)wehave
|Ae−σAx|B(X)σ−1+θ|Aθx|B(X),x∈X.(1.3.13)
UsingHölder’sinequality,(1.3.13),(1.1.17),Hardy’sinequality(Lemma1.1.2)andPropo-
sition1.1.11,weestimatethefirstsummandin(1.3.12)by
∞σp
σ−p(2s+α)τ1−µ|Ae−σAu(τ)|dτdσ
0∞σ0
≤τp(1−µ)|Ae−σAu(τ)|pσp−1σ−p(2s+α)dτdσ
0∞0σ
τp(1−µ)|As+αu(τ)|pσ−(1+ps)dτdσ
0∞0σ
τp(1−µ)|(u(σ)−u(τ))|ps+ασ−(1+ps)dτ+σp(1−µ−s)|u(σ)|ps+αdσ
00D(A)D(A)
p|u|Wsp,µ(R+;D(As+α)).
Wefurtheruse(1.3.13),theHardy-Younginequality(A.2.1),Hölder’sinequalityand
(1.1.17)toestimatethesecondsummandin(1.3.12),
∞σtp
σ2−µ−(2s+α)t−(3−µ)τ1−µ|Ae−σA(u(t)−u(τ))|dτdtdσ
0∞00σt
σ−(s+α−(1−µ+1/p))t−1t−(2−µ)τ1−µ|u(t)−u(τ)|sdτdtpσ−1dσ
000D(A)
∞σp
σ−p(s+α−(1−µ+1/p))σ−p(2−µ)τ1−µ|u(σ)−u(τ)|D(As)dτσ−1dσ
0∞σ0
≤τp(1−µ)|u(σ)−u(τ)|pD(As)σ−(1+p(s+α))dτdσ≤[u]pWsp,µ+α(R+;D(As)),
00whichshows(1.3.9).
Fromtheabovelemmawededuceageneraltracetheoremfortheweightedanisotropic
spaces.Wereferto[89,Theorem3.2.1]fortheunweightedcase,andto[27,Lemma4.4]
foranisotropicspaceswithexponentialweights.
Theorem1.3.6.LetEbeaBanachspaceofclassHT,letJ=(0,T)befiniteorinfinite,
andletΩ⊂Rnbeaboundeddomainwithsmoothboundary,orΩ∈{Rn,R+n}.Assume
r≥0,β>0,andsupposethatk∈N0,s≥0,andα∈(0,2)satisfy
k−µ+1/p<s<k+1−µ+1/p<s+α.

40

TheSpacesLp,µandWeightedAnisotropicSpaces

SetHsp,µ(Wpr):=Hsp,µ(J;Wpr(Ω;E)),andanalogouslyfortheotheranisotropicspaces.
Throughout,assumethattheordersofdifferentiabilityofalloccurringWp,µ-andW-spaces
arenoninteger.Theneachofthespaces
Hsp,µ+α(Wpr)∩Hsp,µ(Wpr+β),Wsp,µ+α(Hpr)∩Wsp,µ(Hpr+β),Wsp,µ+α(Hpr)∩Hsp,µ(Wpr+β),
(1.3.14)iscontinuouslyembeddedinto
BUCkJ,Brp,p+β(1+(s−(k+1−µ+1/p))/α)(Ω;E).(1.3.15)
Moreover,forα≤1itholds
Wαp,µ(Wpr)∩Lp,µ(Wpr+β)→BUCJ,Brp,p+β(1−(1−µ+1/p)/α)(Ω;E),(1.3.16)
W1p,µ(Wpr)∩Wsp,µ(Wpr+β)→BUCJ,Brp,p+β(µ−1/p)/(1−s)(Ω;E).(1.3.17)
AlltheseembeddingsremaintrueifonereplacesΩbyitsboundary∂Ω.Restrictingto
s+α≤2and0Hp,µ-resp.0Wp,µ-spacesintime,theembeddingconstantsareindependent
ofthelengthofJ.
Proof.(I)Usingextensionsandrestrictions,itagainsufficestotreatthecaseJ×Ω=
R+×Rn.Weonlyhavetoconsiderthecasek=0,sincefork≥1itholds,dueto
s>k−µ+1/p,(1.1.1)andProposition1.1.11,
Hsp,µ(Hpr+β)∩Hsp,µ(Wpr+β)∩Wsp,µ(Hpr+β)→BUCk−1(J;Hpr+β),
andthelatterspaceembedsinto(1.3.15).Wefurtherclaimthattheproofoftheasserted
embeddingreducestoshowthatthetemporaltraceoperatortr0u=u(0)mapseachofthe
fivespacesunderconsiderationcontinuouslyinto
Y:=Brp,p+β(1+(s−(k+1−µ+1/p))/α)(Rn;E),
whereonehastosets=k=0for(1.3.16)andk=0,α=1−s,for(1.3.17).Toseethis,
notethatforafunctionuwehave
u(t)=tr0Λtu,t≥0,
whereΛtdenotesthelefttranslationbyt.DuetoLemma1.1.6,thefamilyoflefttransla-
tionsformsoneachspaceWκp,µ,Hκp,µ,κ≥0,astronglycontinuoussemigroupofcontrac-
tions.Wethushavefort>τ≥0,assumingthattr0iscontinuous,
|u(t)−u(τ)|Y|Λt−τu−u|Y,
whereYstandsforanyofthespacesunderconsideration.Thisshowsuniformcontinuity
andboundednessofuwithvaluesinY.
(II)Weshowtheassertedcontinuityoftr0onthespacespacesin(1.3.14).Itfollowsfrom
that1.3.2sitionoPropHsp,µ+α(Wpr)∩Hsp,µ(Wpr+β)→Wsp,µ+(1−ε)α(Hpr+εβ)∩Wsp,µ+εα(Hrp+(1−ε)β).

SpacesAnisotropiceightedW1.3

41

Wsp,µ+α(Hpr)∩Hsp,µ(Wpr+β)→Wsp,µ+(1−ε)α(Hpr+εβ)∩Wsp,µ+εα(Hpr+(1−ε)β),
whereε>0issufficientlysmall.Sinceitisassertedthatthespacesontheleft-andthe
right-handsideabovehavethesametracespaces,itsufficestoconsidertr0on
Wsp,µ+α(Hpr)∩Wsp,µ(Hpr+β).
Moreover,usingagainProposition1.3.2,thesameargumentshowsthatitsufficestocon-
siderthecases+α<1.Weapply(1.3.9)with
X=Hpr−sβ/α,A=(1−Δn)β/2α,D(A)=Hpr+(1−s)β/α.
Sincetheoperator1−ΔnadmitsaboundedH∞-calculuswithH∞-angleequaltozeroon
thewholeH-scale,itfollowsfrom(A.3.1),LemmaA.3.5andPropositionA.4.2that
D(As)=Hpr,D(As+α)=Hpr+β.
ThusLemma1.3.5impliesthattr0mapscontinuously
Wsp,µ+α(Hpr)∩Wsp,µ(Hpr+β)→DA2s+α−(1−µ+1/p),p=Brp,p+β(1+(s−(k+1−µ+1/p))/α).
(III)Itfollowsfromrealinterpolationthattheoperator1−Δnhasthesameproperties
ontheB-scaleasontheH-scale.Wemaythereforeuse(1.3.10),appliedto
X=Brp,p,A=(1−Δn)β/2α,D(A)=Brp,p+β/α,
givingDA(α−(1−µ+1/p),p)=Brp,p+β(1−(1−µ+1/p)/α),toobtain(1.3.16).Similarly,applying
with(1.3.11)X=Brp,p−sβ/(1−s),A=(1−Δn)β/2(1−s),D(A)=Wpr+β,
givingDA(1+s−(1−µ+1/p),p)=Brp,p+β(µ−1/p)/(1−s),yields(1.3.17).
TheabovetheoremcanagainbevisualizedbytheNewtonpolygon,cf.Figure1.3.2.Con-
sider,forinstance,thespaceX=Wsp,µ+α(Hrp)∩Wsp,µ(Hpr+β),wheres,r,αandβareas
above.ThenthetemporaltracespaceofXisobtainedbyintersectingthehorizontalline
(τ,1−µ+1/p),τ∈R,withthenontrivialpartoftheNewtonpolygonNPcorresponding
.XtoRemark1.3.7.InthesituationofTheorem1.3.6,onecanalsoconsiderthecase
k1−µ+1/p<s<k2+1−µ+1/p<s+α,0≤k1≤k2,k1,k2∈N0.
Thiscasecanbereducedtok1=k2,wherethetheoremisapplicable,usingProposition
1.3.2.Hereonehasthechoicebetweenhightemporalandlowspatialregularityandvice
ersa.vUsingargumentsasintheproofof[27,Theorem4.5],oneshouldbeabletoshowthatthe
temporaltraceissurjective,forallofthespacesunderconsiderationintheTheorem1.3.6.
Atthispointweonlyconsideraright-inverseinaspecialcase.WealsorefertoLemma
3.2.2,whereweconsideraright-inversefortheboundaryspacesfromChapter3.

42

TheSpacesLp,µandWeightedAnisotropicSpaces

(τ,1−µ+1/p)

spatialregularityofthetemporaltrace

Figure1.2:ThetracespaceandtheNewtonpolygon

Lemma1.3.8.LetEbeaBanachspace,letp∈(1,∞),µ∈(1/p,1],andlet−Abethe
generatorofanexponentiallystableanalyticC0-semigrouponE,withdomainD(A).Let
furtherα>1−µ+1/pwithα−(1−µ+1/p)∈/N.Thenitholds
|e−∙Ax|Wαp,µ(R+;E)∩Lp,µ(R+;DA(α,p))|x|DA(α−(1−µ+1/p),p).9
Proof.(I)Firstletα∈N.Using(1.3.8),forx∈DA(α−(1−µ+1/p),p)weobtain
|e−∙Ax|Lp,µ(R+;D(Aα))=|Ae∙AAα−1x|Lp,µ(R+;E)
|Aα−1x|DA(µ−1/p,p)=|x|DA(α−(1−µ+1/p),p).
Since∂tke−∙Ax=(−A)ke−∙Axfork≤α,wefurtherhave
|e−∙Ax|Wαp,µ(R+;E)|e−∙Ax|Lp,µ(R+;D(Aα)),
whichshowstheassertionforintegerα.
(II)Wenowconsiderthecase1−µ+1/p<α<1,andshowe−∙Ax∈Lp,µ(R+;DA(α,p)).
Takex∈DAα−(1−µ+1/p),p.Thenitholds|e−∙Ax|Lp,µ(R+;E)|x|E,duetothe
exponentialstabilityofthesemigroup.Moreoverwehave
|e−∙Ax|Lpp,µ(R+;D(α,p))=∞∞sp(1−µ)tp(1−α)|Ae−(t+s)Ax|Epdsdt.
tA00Wesplittheinnerintegralats=tandestimatethefirstsummandwithsomesmallε>0
yb

t∞sp(1−µ)tp(1−α)|Ae−(t+s)Ax|pdsdt
E00∞tt
tp(1−α+(1−µ+1/p))|Ae−tAx|p1e−εpsdsdt
Ett00p[x]DA(α−(1−µ+1/p),p).
9RecallthatDA(α,p)=D(Aα)forα∈N0.

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43

ForthesecondsummandweusetheHardy-Younginequality(A.2.2),toestimate
∞∞sp(1−µ)tp(1−α)|Ae−(t+s)Ax|Epdsdt
0tt
∞tp(1−α)∞sp(1−µ+1)|Ae−sAx|pdsdt
∞0tEst
sp(1−α+(1−µ+1/p))|Ae−sAx|pds
Es0p=[x]DA(α−(1−µ+1/p),p).
Wethusobtainfor1−µ+1/p<α<1that
|e−∙Ax|Lp,µ(R+;DA(α,p))|x|DA(α−(1−µ+1/p),p).(1.3.18)
Nowletα>1withα∈/N.Thenwechooseβ>0suchthat1−µ+1/p<α−β<1,
whichyieldsAβx∈DA(α−β−(1−µ+1/p),p)bythereiterationtheorem.Itnowfollows
that(1.3.18)from|e−∙Ax|Lp,µ(R+;DA(α,p))|e−∙AAβx|Lp,µ(R+;DA(α−β,p))
β|Ax|Lp,µ(R+;DA(α−β−(1−µ+1/p),p))|x|Lp,µ(R+;DA(α−(1−µ+1/p),p)),
andtherefore(1.3.18)holdsforallα>1−µ+1/p.
(III)Forthetemporalregularity,observethat
∂tke−∙Ax=(−A)ke−∙Ax,k∈{0,...,[α]},
wherex∈DA(α−(1−µ+1/p),p).Inviewoftheexponentialstabilityofthesemigroup,
therepresentations(1.1.18)forthenormofWθp,µ(R+;E)andA.2l)forthenormofDA(α−
(1−µ+1/p),p)wehave
∞∞dt
|Ake−∙Ax|pWθ(R+;E)∼|Ake−∙Ax|Ep+sp(1−µ)t−θp|Ake−(t+s)Ax−AkesAx|Epds
tp,µ00∼|Ake−∙Ax|Lpp,µ(R+;DA(θ,p))
fork=[α]andθ∈(0,1).Thisyieldsthatforα>1−µ+1/pwithα∈/N,using(1.1.8)
andtheestimatesofStepII,
[α][α]
|e−∙Ax|Wαp,µ(R+;E)|∂tke−∙Ax|Lp,µ(R+;E)+|∂te−∙Ax|Wαp,µ−[α](R+;E)
=0kA−∙|ex|Lp,µ(R+;DA(α,p))|x|DA(α−(1−µ+1/p),p),
whichfinishestheproof.
Animmediateapplicationoftheaboveresultyieldsacontinuousright-inverseofthetempo-
raltraceforweightedanisotropicspacesarisinginthecontextofmaximalLp,µ-regularity.

44

TheSpacesLp,µandWeightedAnisotropicSpaces

Lemma1.3.9.LetEbeaBanachspaceofclassHT,p∈(1,∞),µ∈(1/p,1],m∈N,
andletΩ⊂Rnbeadomainwithcompactsmoothboundary∂Ω.Thenaright-inversefor
tr0thatiscontinuous
B2p,pm(µ−1/p)(Ω;E)→W1p,µ(R+;Lp(Ω;E))∩Lp,µ(R+;Wp2m(Ω;E))
ybengivist→RRne−t(1−Δn)mEΩu0,u0∈B2p,pm(µ−1/p)(Ω;E).
HereEΩistheextensionoperatortoRnfrom(1.3.2),andRRndenotestherestrictionfrom
n.ΩtoR

racesTSpatial1.3.3Wenowspecializetoweightedanisotropicspacesoftheform
Hs,p,µ2ms(J×Ω;E):=Hsp,µ(J;Lp(Ω;E))∩Lp,µ(J;Hp2ms(Ω;E)),(1.3.19)
wherem∈Nands∈(0,1],andtothecorrespondingspaceswhereHisreplacedbyW.
Ourmotivationistoinvestigatethemappingpropertiesofaboundarydifferentialoperator
trΩβwithβ∈N0nand|β|≤2m−1,wheretrΩanddenotethespatialtrace,i.e.,
trΩu=u|∂Ω,andtheeuclidiangradientonRn,respectively.Theiterativeapplicationof
Lemma1.3.4impliesthatβmapsthemaximalregularityspace
W1p,µ(J;Lp(Ω;E))∩Lp,µ(J;Wp2m(Ω;E))
continuouslyinto
H1p,µ−|β|/2m(J;Lp(Ω;E))∩Lp,µ(J;Hp2m−|β|(Ω;E)),
whichisaspaceasin(1.3.19)withs=1−|β|/2m.Wearethereforeledtoinvestigatethe
propertiesoftrΩonaspacelike(1.3.19).Wefollowtheproofof[25,Lemma3.5].Forthe
spatialtraceonunweightedanisotropicspaceswealsoreferto[11,Chapter4].
Forourfurtherconsiderationsweassumethat
.Nms2∈ItisknownthatthetraceoperatortrΩ,whichisoriginallyonlydefinedonCc∞(Rn;E),
extendsuniquelytoacontinuousmap
Hp2ms(Ω;E)→Wp2ms−1/p(∂Ω;E).(1.3.20)
Thiscanbeseenasin[82,Theorems2.9.3,4.7.1]forthescalar-valuedcase.Applied
pointwisealmosteverywhereintime,trΩextendsfurthertoacontinuousmap
Lp,µ(J;Hp2ms(Ω;E))→Lp,µ(J;Wp2ms−1/p(∂Ω;E)).
ObservethatProposition1.3.2yieldstheembedding
Hs,p,µ2ms(J×Ω;E)→Hsp,µ−1/2mp(J;H1p/p(Ω;E)).

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45

AlthoughtrΩisnotcontinuousfromHp1/p(Ω;E)toLp(∂Ω;E),thisfactsuggeststhattrΩ
mapsHs,p,µ2ms(J×Ω;E)intoHsp,µ−1/2mp(J;Lp(∂Ω;E)).
Togivearigorousproof,thefollowingsimpledensityresultisuseful.
Lemma1.3.10.LetJ=(0,T)beafiniteorinfiniteinterval,letEbeaBanachspace,
andletDbeadensesubsetofE.ThenthesetStep(J;D),consistingstepfunctionsofthe
formlφ=αi(∙)φi,αi∈Cc(J),φi∈D,l∈N,
=1iisdenseinLp,µ(J;E).
Proof.SinceCc(J;E)isdenseinLp,µ(J;E),itsufficestoapproximatefunctionsfromthis
set.Letε>0begivenandtakeu∈Cc(J;E),suchthatsuppu⊂(a,b)forsomea,b∈J.
Choosenumbersa=t1<...<tl−1<tl=b,l∈N,with
|u(t)−u(ti)|E<εfort∈[ti,ti+1],i=1,...,l.
Byassumption,foreachithereisϕi∈Dsuchthat|u(ti)−ϕi|E<ε,wherewecantake
ϕ1=ϕl=0.Nowdefineφ∈Step(J;D)by
lφ(t)=1[ti,ti+1)(t)(ti+1−t)ϕi+(t−ti)ϕi+1,t∈J.
i=1ti+1−ti
Then|u−φ|L∞(J;E)<2ε,andthus
|u−φ|Lp,µ(J;E)<2bp(1−µ)(b−a)ε.
Sinceaandbonlydependonu,theassertionfollows.
Letusnowassumethat
J×Ω=R+×R+n.
ForthiscasewedescribeanalternativerepresentationoftrR+n.Inthesequelwewrite
x=(x,y)∈R+n,x∈Rn−1,y∈R+.
Consideringafunctionu=u(t,x,y)onR+×R+nasafunctionofy∈R+withvaluesin
thefunctionsof(t,x)∈R+×Rn−1,Fubini’stheoremyieldstheembedding
ι1:Lp,µR+;Hp2ms(R+n;E)→Hp2msR+;Lp,µ(R+;Lp(Rn−1;E)).
Thus,since2ms≥1,thetracetr0:=try=0actsonLp,µ(R+;Hp2mτ(R+n;E))viatr0◦ι1,and
mapsthisspacecontinuouslyintoLp,µ(R+;Lp(Rn−1;E)).Forφ∈Step(R+;Cc∞(Rn;E))
ittriviallyholdstrR+nφ=(tr0◦ι1)φ.DuetothedensityofStep(R+;Cc∞(Rn;E)),proved
inLemma1.3.10,weobtainthat
trR+n=tr0◦ι1onLp,µ(R+;Hp2ms(R+n;E)).(1.3.21)
Thisrepresentationallowstoprovethetemporalregularityforspatialtracesoffunctions
inHs,p,µ2ms(R+×R+n;E)assuggestedabove.

46

TheSpacesLp,µandWeightedAnisotropicSpaces

Lemma1.3.11.LetEbeaBanachspaceofclassHT,andletm∈Nands∈(0,1]
satisfy2ms∈N.ThenthetracetrR+nmapscontinuously
Hs,p,µ2ms(R+×R+n;E)→Wsp,µ−1/(2mp),2ms−1/p(R+×Rn−1;E).
Proof.ThroughoutthisproofwesetX:=Lp,µ(R+;Lp(Rn−1;E)).
(I)ConsideringafunctioninLp,µ(R+;H2pms(R+n;E))asafunctionofy∈R+takingvalues
inthefunctionsof(t,x)∈R+×Rn−1,weobtainthat
Lp,µ(R+;Hp2ms(R+n;E))→Hp2ms(R+;X).
Moreover,itfollowsfromHp2ms(R+n;E)→Lp(R+;Hp2ms(Rn−1;E))andFubini’stheorem
thatLp,µ(R+;Hp2ms(R+n;E))→Lp(R+;Lp,µ(R+;Hp2ms(Rn−1;E))).
Fubini’stheoremandinterpolationfurtheryield
Hsp,µ(R+;Lp(R+n;E))=Lp(R+;Hsp,µ(R+;Lp(Rn−1;E))).
ByLemma1.3.1,therealizationoftheoperatorL=1−∂t+(−Δn−1)monXisinvertible
andadmitsboundedimaginarypowerswithpoweranglenotexceedingπ/2.Hence,by
LemmaA.3.5,forτ∈(0,1]itspowerLτhasboundedimaginarypowerswithanglenot
largerthanτπ/2,anditholds
D(Lτ)=Hτp,µ(R+;Lp(Rn−1;E))∩Lp,µ(R+;Hp2mτ(Rn−1;E)).(1.3.22)
ThereforeHs,p,µ2ms(R+×R+n;E)→Hp2ms(R+;X)∩Lp(R+;D(Ls)).
Denotingtheaboveembeddingbyι1,equation(1.3.21)impliestrR+n=tr0◦ι1.
(II)WenowclaimthatthespaceHp2ms(R;X)∩Lp(R;D(Ls))embedscontinuouslyinto
Hp1(R;D(Ls−1/2m)).Toseethis,weconsidertherealizationoftheoperatorsA=1+
(−∂y2)smandB=LsonLp(R;X)withdomains
D(A)=Hp2ms(R;X)andD(B)=Lp(R;D(Ls)),
respectively.Theseoperatorsareinvertible,andadmitboundedimaginarypowerswith
poweranglesequaltozeroandsπ/2,respectively.Moreover,AandBareresolventcom-
mutingonstepfunctionsinLp(R;X),whichcarriesovertoLp(R;X)bydensity.Thusthe
Dore-VenniTheoremA.3.2showsthattheoperatorA+BisinvertibleonLp(R;X)with
domainD(A+B)=Hp2ms(R;X)∩Lp(R;D(Ls)).
Sinceitholdsthat|A1/2msB1−1/2ms∙|Lp(R;X)and|(A+B)∙|Lp(R;X)areequivalentnorms
onHp1(R;D(Ls−1/2m))andD(A+B),respectively,themixedderivativetheorem(Lemma
A.3.3)impliestheassertedembedding.
(III)Itfollowsfromrestrictionandextensionthatalso
Hp2ms(R+;X)∩Lp(R+;D(Ls))→Hp1(R+;D(Ls−1/2m)),

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47

whichimpliesthattheoperatorLs−1/2mmapscontinuously
Hp2ms(R+;X)∩Lp(R+;D(Ls))→Hp1(R+;X)∩Lp(R+;D(L1/2m)).
NotethatL1/2missectorialofangleatmostπ/4m<π/2,andthus−L1/2misthegenerator
ofanexponentiallystableanalyticC0-semigrouponX.Dueto[7,TheoremIII.4.10.2]we
evhaHp1(R+;X)∩Lp(R+;D(L1/2m))→BUC([0,∞);DL1/2m(1−1/p,p)),
andfromthereiterationtheoremweinfer
DL1/2m(1−1/p,p)=DL((1−1/p)/2m,p).
(IV)Wenowwrite
trR+n=tr0L−(s−1/2m)Ls−1/2mι1,
whereLs−1/2manditsinverseareappliedpointwise.Bytheaboveconsiderations,the
operatorLs−1/2mι1mapscontinuously
Hs,p,µ2ms(R+×R+n;E)→BUC([0,∞);DL((1−1/p)/2m,p)).
Clearly,tr0andL−(s−1/2m)commuteonBUC([0,∞);DL((1−1/p)/2m,p)),andbyreit-
erationandLemmaA.3.1,L−(s−1/2m)mapsDL((1−1/p)/2m,p)continuouslyinto
DL(s−1/2mp,p)=Wsp,µ−1/(2mp),2ms−1/p(R+×Rn−1;E).
ThisshowsthatthetracetrR+nmapscontinuouslyasasserted.
Vialocalizationweextendtheaboveresulttogeneraldomainsandfiniteintervals.
Proposition1.3.12.LetEbeaBanachspaceofclassHT,p∈(1,∞),µ∈(1/p,1],let
m∈Nands∈(0,1]besuchthat2ms∈N,letJ=(0,T)beafiniteorinfiniteinterval,
andletΩ⊂Rnbeadomainwithcompactsmoothboundary,orΩ∈{Rn,R+n}.Thenthe
spatialtracetrΩmapscontinuously
Hs,p,µ2ms(J×Ω;E)→Wsp,µ−1/2mp,2ms−1/p(J×∂Ω;E).
TheoperatornormoftrΩon0Hτp,µ,2mτ(J×Ω;E)isindependentofthelengthofJ.
Proof.(I)UsingtheextensionoperatorsEJandEJ0fromLemma1.1.5,itsufficesto
considerthecaseJ=R+.Wedescribe∂Ωbyafinitenumberofcharts(Ui,ϕi)anda
partitionofunity{ψi}subordinatetothecoveriUi.WefurtherdenotebyΦithepush-
forwardwithrespecttoϕi,i.e.,Φiu=u◦ϕi−1.Forafunctionφ∈Step(R+;Cc∞(Rn;E))
holdsittrΩφ=Φi−1trR+nΦi(ψiφ)on∂Ω.(1.3.23)
i(II)ByrestrictiontoΩ∩Ui,LemmaA.4.1andtrivialextensionfromR+n∩ϕi(Ui)toR+n,
foreachiweobtainthattheΦi(ψi∙)mapscontinuously
Hs,p,µ2ms(R+×Ω;E)→Hs,p,µ2ms(R+×Rn+;E).

48

TheSpacesLp,µandWeightedAnisotropicSpaces

ApplyingLemma1.3.11,restrictingbacktoR+n∩ϕi(Ui)andusingagainLemmaA.4.1
yieldsthatΦi−1trR+nmapsthelatterspacecontinuouslyinto
Wsp,µ−1/2mp,2ms−1/p(R+×∂Ω;E).
ThustheoperatoriΦi−1(trR+nΦi(ψi∙))mapscontinuously
Hs,p,µ2ms(R+×Ω;E)→Wsp,µ−1/2mp,2ms−1/p(R+×∂Ω;E).
SinceStep(R+;Cc∞(Rn;E))isdenseinLp,µ(R+;Hp2ms(Ω;E))byLemma1.3.10,there-
presentation(1.3.23)holdsforallelementsofthisspace,andinparticularforallfunctions
fromHs,p,µ2ms(J×Ω;E).ThisshowsthattrΩiscontinuousasasserted.
Arguingasin[25,Lemma3.5]onecanshowthatinthesituationoftheaboveproposition
thespatialtraceissurjective.
Weusetheresultsderivedsofartoestimatedifferentialoperatorsoflowerorderonspaces
(1.3.19).eyptofLemma1.3.13.LetEbeaBanachspaceofclassHT,p∈(1,∞)andµ∈(1/p,1].Let
J=(0,T)beafiniteinterval,andletΩ⊂Rnbeadomainwithcompactsmoothboundary
∂Ω,orΩ∈{Rn,R+n}.Letfurtherthenumbersm∈Nands∈[0,1)begiven.Thenfor
everyη>0thereisT0>0suchthatforT≤T0thefollowingholdstrue.
a)Forα∈N0nwiths+|α|/2m<1itholds
|αu|0Hs,p,µ2ms(J×Ω;E)≤η|u|W1p,µ,2m(J×Ω;E)foru∈0W1p,µ,2m(J×Ω;E).
b)Forβ∈N0nwiths+|β|/2m+1/2mp<1itholds
|trΩβu|0Ws,p,µ2ms(J×∂Ω;E)≤η|u|W1p,µ,2m(J×∂Ω;E)foru∈0W1p,µ,2m(J×Ω;E).
Proof.(I)ItfollowsfromLemma1.3.4thatthereisconstantC0,whichisindependent
ofJ,suchthat
α|u|0Hs,p,µ2ms(J×Ω;E)≤C0|u|0Hsp,µ+|α|/2m,2ms+|α|(J×Ω;E).
FromtheinterpolationinequalityA.2j),theassumptions+|α|/2m<1andYoung’s
inferewyinequalit|u|0Hsp,µ+|α|/2m(J;Lp(Ω;E))≤|u|0sW+|1αp,µ|/(2J;mLp(Ω;E))|u|L1−p,µs(−|J;αL|/p2(Ω;mE))
η≤4C0|u|0W1p,µ(J;Lp(Ω;E))+Cη|u|Lp,µ(J;Lp(Ω;E)),
whereCηisaconstantthatdependsonη.Hereitisimportantthatforcomplexinterpola-
tiontheconstantintheinterpolationinequalityisequalto1andthusindependentofthe
underlyingspaces.ItfurtherfollowsfromPoincaré’sinequality(Lemma1.1.14)that
η|u|Lp,µ(J;Lp(Ω;E))≤4C0Cη|u|W1p,µ(J;Lp(Ω;E)),

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49

providedthatT≤T0withsufficientlysmallT0.Thisshows
η|u|0Hsp,µ+|α|/2m(J;Lp(Ω;E))≤2C0|u|W1p,µ(J;Lp(Ω;E)).
Inasimilarwayweestimate
η|u|Lp,µ(J;Hp2ms+|α|(Ω;E))≤4C0|u|Lp,µ(J;Wp2m(Ω;E))+Cη|u|Lp,µ(J;Lp(Ω;E))
η≤2C0|u|Lp,µ(J;Wp2m(Ω;E))+|u|W1p,µ(J;Lp(Ω;E)).
a).wsshoThis(II)Forb)weobtainasabovethatforgivenη>0itholds
|trΩβu|0Ws,p,µ2ms(J×∂Ω;E)≤η|trΩβu|0W1p,µ−|β|/2m−1/2mp,2m−|β|−1/p(J×∂Ω;E),
providedthatT≤T0issufficientlysmall.ByProposition1.3.12andLemma1.3.4there
isconstantC0,whichdoesnotdependonJ,suchthat
|trΩβu|0W1p,µ−|β|/2m−1/2mp,2m−|β|−1/p(J×∂Ω;E)≤C0|u|W1p,µ,2m(J×Ω;E).
Settingη=η/C0,weobtaintheassertedestimate.
Weendthissectionwithausefuldensityresultforanisotropicspaces.
Lemma1.3.14.LetEbeaBanachspaceofclassHT,p∈(1,∞),µ∈(1/p,1],andlet
s=1ors∈(0,1)with2ms∈/Nands=1−µ+1/p.Then
Cc∞(R+;Wp2m(Rn;E))→d0Ws,p,µ2ms(R+×Rn;E).
Proof.ThroughoutwesetY0:=Lp(Rn,E),andY1:=Wp2m(Rn,E).
(I)Wefirstconsiderthecases=1,i.e.,weshowthatCc∞(R+;Y1)isdensein0W1p,µ,2m(R+×
Rn;E).Tothisendwefirstshowthatthesetoffunctionsin0W1p,µ,2m(R+×Rn;E)which
arecompactlysupportedinR+aredensein0W1p,µ,2m(R+×Rn;E).Letε>0andu∈
0W1p,µ,2m(R+×Rn;E)begiven.ChooseTε>tε>0suchthatthenumbers
|1R+\(tε,Tε)u|Lp,µ(R+;Y0),|1R+\(tε,Tε)u|Lp,µ(R+;Y0),|1R+\(tε,Tε)u|Lp,µ(R+;Y1),
aresmallerthanε,respectively.Choosefurtherasmoothnonnegativecut-offfunctionαε
onR+withαε≤1and
1,t∈(tε,Tε),
αε(t)=0,t∈(0,tε/2),|αε||(tε/2,tε)t1,|αε||(Tε,Tε+1)1.
0,(Tε+1,∞),ε
holdsitThen|u−αεu|Lp,µ(R+;Y0)ε,|u−αεu|Lp,µ(R+;Y1)ε,
thatfurtherand|u−(αεu)|Lp,µ(R+;Y0)≤|u−αεu|Lp,µ(R+;Y0)+|αεu|Lp,µ(R+;Y0)
ε+|αεu|Lp,µ(R+;Y0).

50

TheSpacesLp,µandWeightedAnisotropicSpaces

Thepropertiesofαεyield
|αεu|Lp,µ(R+;Y0)|αεu|Lp,µ(tε/2,tε;Y0)+|αεu|Lp,µ(Tε,Tε+1;Y0).(1.3.24)
Toestimatethefirstsummandweusethat1/tεp≤1/tpfort≤tε,toobtain
tpε|αεu|Lpp,µ(tε/2,tε;Y0)≤t2εp|u|Lpp,µ(tε/2,tε;Y0)t−pµ|u(t)|Yp0dtε
2/tεfortεsufficientlysmall,sincethefunctiont−pµ|u|Yp0belongstoL1(R+)byHardy’sinequal-
ity(Lemma1.1.2).Theassumptionforαεon(Tε,Tε+1)impliesthatthesecondsummand
of(1.3.24)issmallerthanaconstantmultipleofε.Hence|αεu|Lp,µ(R+;Y0)ε,andthus
|u−αεu|W1p,µ(R+;Y0)+|u−αεu|Lp,µ(R+;Y1)ε.
Thereforethefunctionsαεu,whichbelongtoW1p,µ,2m(R+×Rn;E)andaresupportedin
(tε/2,Tε+1),approximateuinW1p,µ,2m(R+×Rn;E)asε0.
(II)Toapproximateafunctionu∈0W1p,µ,2m(R+×Rn,E)withcompactsupportinR+
byfunctionsinCc∞(R+;Y1)wehavetoapproximateuinWp1(R+;Y0)andLp(R+;Y1)
simultaneously.Thiscanbeachievedusingastandardmollificationmethod,asinthe
proofsof[1,Theorem2.29,Lemma3.16].Weomitthedetails.Theassertionofthislemma
fors=1follows.
(III)Nowlets∈(0,1).ByA.2a)thedenseembedding
0W1p,µ,2m(R+×Rn;E)→d(Lp,µ(R+;Y0),0W1p,µ,2m(R+×Rn;E))s,p
isvalid,andtheLemmas1.1.9,1.3.1andPropositionA.4.2yield
(Lp,µ(R+;Y0),0W1p,µ,2m(R+×Rn;E))s,p=0Ws,p,µ2ms(R+×Rn;E),
provided2ms∈/Nands=1−µ+1/p.

1.3.4PointwiseMultipliers
IfE,J,andΩareasinLemma1.3.4,thentheoperatorα,whereα∈N0n,|α|≤2mand
m∈N,mapscontinuously
W1p,µ,2m(J×Ω;E)→H1p,µ−|α|/2m,2m−|α|(J×Ω;E).10
Motivatedbylineardifferentialoperatorswithvariablecoefficients,wearelookingfor
sufficientconditionsonafunctiona=a(t,x)∈B(E)tobeapointwisemultiplierto
Lp,µ(J;Lp(Ω;E)),i.e.,suchthatthemultiplicationwithitisacontinuousmap
Hτp,µ,2mτ(J×Ω;E)→Lp,µ(J;Lp(Ω;E)),τ∈(0,1].
Wehavethefollowingresultforcoefficientsawhichbelongtoanunweightedspace.
10Recallforτ>0thenotationHτp,µ,2mτ(J×Ω;E)=Hτp,µ(J;Lp(Ω;E))∩Lp,µ(J;Hp2mτ(Ω;E)),and
-spaces.Wtheforanalogously

SpacesAnisotropiceightedW1.3

51

Lemma1.3.15.LetEbeaBanachspaceofclassHT,letJ=(0,T)befinite,andlet
Ω⊂Rnbeadomainwithcompactsmoothboundary,orletΩ∈{Rn,R+n}.Letfurther
p∈(1,∞),µ∈(1/p,1],r,s∈[p,∞),andτ∈(0,1]satisfy
p(1−µ)+1+n<τ.
mr2sThenthereisC>0suchthat
|au|Lp,µ(J;Lp(Ω;E))≤C|a|Ls(J;Lr(Ω;B(E)))|u|Hτp,µ,2mτ(J×Ω;E)
isvalidforalla∈Ls(J;Lr(Ω;B(E)))andu∈Hτp,µ,2mτ(J×Ω;E).Restrictingtou∈
0Hτp,µ,2mτ(J×Ω;E),forgivenT0>0theconstantCmaybechosenuniformlyforall
.TT0≤Proof.ApplyingHölder’sinequalitytwiceyields
|au|Lpp,µ(J;Lp(Ω;E))=tp(1−µ)|a(t,∙)u(t,∙)|Lpp(Ω;E)dt
J≤|a(t,∙)|Lpr(Ω;B(E))|t1−µu(t,∙)|Lpr(Ω;E)dt
Jpp≤|a|Ls(J;Lr(Ω;B(E)))|u|Ls,µ(Lr(Ω;E)),
wherer1+r1=s1+s1=p1.11DuetoProposition1.3.2,forσ∈(0,1)theembedding
Hτp,µ,2mτ(J×Ω;E)→Hτp,µ(1−σ)(J;Hp2mτσ(Ω;E)),
isvalid,andtheembeddingconstantisindependentofJifonerestrictsto0Hp,µ-spaces
intime.Sobolev’sembeddingyields
Hp2mτσ(Ω;E)→Lr(Ω;E)forσ=2mτnr<1.
ItfollowsfromProposition1.1.12that
Hτp,µ−2nmr(J;Lr(Ω;E))→Ls(J;Lr(Ω;E))forτ−n−1−µ+1>−p(1−µ+1/p),
spmr2withanembeddingconstantasassertedinthe0Hp,µ-case.Sincethelatterconditionis
equivalenttop(1−µs+1/p)+2nmr<τ,thisfinishestheproof.
Wearealsointerestedinthecasewherethecoefficientsbelongtoatemporallyweighted
space.If2m(µ−1/p)>2m−1+n/p,thenTheorem1.3.6andSobolev’sembeddingyield
W1p,µ,2m(J×Ω;E)→C(J;BUC2m−1(Ω;E)).
Thusαmapsfor|α|<2mcontinuously
W1p,µ,2m(J×Ω;E)→BUC(J×Ω;E),
andthemultiplicationwithaiscontinuousfromBUC(J×Ω;E)toLp,µ(J;Lp(Ω;E))
ifa∈Lp,µ(J;Lp(Ω;B(E))).TheseconsiderationstogetherwithLemma1.3.15yieldthe
followingresultfordifferentialoperatorswithvariablecoefficients.
11Notethatrisnotthestandarddualexponentofr.

52

TheSpacesLp,µandWeightedAnisotropicSpaces

Proposition1.3.16.LetEbeofclassHT,p∈(1,∞)andµ∈(1/p,1],letJ=(0,T)
beafiniteinterval,andletΩ⊂Rnbeadomainwithcompactsmoothboundary,orlet
Ω∈{Rn,R+n}.AssumethatfortheB(E)-valuedcoefficienta=a(t,x)oftheoperator
aα,whereα∈N0nwith|α|≤2m,m∈N,itholdsa∈BC(J×Ω;B(E))incase|α|=2m,
andthatincase|α|<2moneofthefollowingconditionsisvalid:either
2m(µ−1/p)>2m−1+n/panda∈Lp,µJ;Lp(Ω;B(E)),
ora∈LsαJ;(Lrα+L∞)(Ω;B(E))forsomenumberssα,rα∈[p,∞)with
p(1−µ)+1+n<1−|α|.
sα2mrα2m
Thenwehave
aα∈BW1p,µ,2m(J×Ω;E),Lp,µ(J;Lp(Ω;E)).
Inthesamesettingasabove,wenowconsiderpointwisemultipliersforanisotropicspaces
relatedtoboundarydifferentialoperators.Lemma1.3.4,togetherwithProposition1.3.11,
yieldsthattrΩβ,where,β∈Nn0,|α|≤2m−1,mapscontinuously
W1p,µ,2m(J×Ω;E)→W1p,µ−|β|/2m−1/2mp,2m−|β|−1/p(J×∂Ω;E).
ThusouraimistoprovidesufficientconditionsonaB(E)-valuedfunctionbsuchthatthe
multiplicationwithitisaboundedlinearmap
Wτp,µ,2mτ(J×∂Ω;E)→Wp,µκ,2mκ(J×∂Ω;E),
where0<κ≤τ<2.Moreover,theestimatesshouldbesuitableforthelocalization
proceduresinthenextchapters.Toobtainrathersharpresultsweusetheparaproduct
techniquespresentedin[74,Section4.4].
Chooseafunctionψ∈Cc∞(Rn)withtheproperty
ψ(ξ)=1,|ξ|≤1,ψ(ξ)=0,|ξ|≥3/2,
anddefinethefamilyϕj,j∈N0,by
ϕ0(ξ)=ψ(ξ),ϕ1(ξ)=ψ(ξ/2)−ψ(ξ),ϕj(ξ)=ϕ1(2−j+1ξ),j≥2.
Thenitholdsj∞=0ϕj≡1onRn,andfurther
ksuppϕj⊂{2j−1≤|ξ|≤32j−1},j∈N,ϕj(ξ)=ψ(2−kξ),k∈N0.
=0jDenotingbyFtheFouriertransformonRn,weusethisdyadicpartitionofunitytodefine
operatorsSjandSkwhichcutoffdyadicfrequenciesintheFourierimage,
kSj:=F−1ϕjF,j∈N0,Sk:=Sj,k∈N0,S−l=S−l:=0,l∈N.
=0j

1.3WeightedAnisotropicSpaces53
Observethatforu∈S(Rn;E)=BS(Rn),Eitholdsu=limk→∞Skuinthesense
ofdistributions.12TheBesovspacesmaybecharacterizedwiththehelpoftheoperators
Sj.By[75,Definition4.3],forσ>0,p∈[1,∞)andq∈[1,∞]theLittlewood-Paley
tationrepresenBσp,q(Rn;E)=u∈S(Rn;E):|u|Bσp,q(Rn;E)=|(2σj|Sju|Lp(Rn;E))j∈N0|lq<∞
isvalid.Herelqdenotethestandardsequencespaces,q∈[1,∞].Weobservethefollowing.
Lemma1.3.17.Fnorq∈[1,∞],theoperatorfamilies(Sj)j∈N0and(Sk)k∈N0areuniformly
boundedonLq(R;E).
Proof.Sinceϕj(ξ)=ψ(2−jξ)−ψ(2−j+1ξ)forj≥1andjk=0ϕj(ξ)=ψ(2−kξ),we
onlyhavetoshowthattheoperatornormoftheconvolutionoperatorF−1ψ(2−j∙)F=
(F−1ψ(2−j∙))∗isboundedindependentofj∈N0.Theconvolutioninequalityshowsthat
forq∈[1,∞]wehave
|(F−1ψ(2−j∙))∗|B(Lq(Rn;E))≤|(F−1ψ(2−j∙))|L1(Rn;E).
NowitiseasytoseethatF−1ψ(2−j∙)(x)=2jn(F−1ψ)(2jx)forx∈Rn,andfurtherthat
|(F−1ψ)(2j∙)|L1(Rn;E)≤2−jn|F−1ψ|L1(Rn;E),
whichyieldsanestimateindependentofj.
Forf∈S(Rn;B(E))∩L1,loc(Rn;B(E))andg∈S(Rn;E)∩L1,loc(Rn;E)weformally
decomposetheproductfgintotheparaproducts
∞∞Π1(f,g):=Sk−2fSkg,Π2(f,g):=(Sk−1f+Skf+Sk+1f)Skg
=0k=2k∞Π3(f,g):=SkfSk−2g,
=2kholdsitthatsofg=Π1(f,g)+Π2(f,g)+Π3(f,g),(1.3.25)
whenevertheparaproductsexistinthesenseofdistributions.Observethatfork∈N0it
holds+1ksuppF(Sk−2fSkg)∪suppFSlfSkg∪suppF(SkfSk−2g)⊂{|ξ|≤2k+3}.
1−k=lThefollowinglemmaisthevector-valuedversionof[74,Proposition2.3.2/2],andgivesa
criterionfortheexistenceofaparaproductinaBesovspace.
12Wereferto[75]and[7]fordetailsonvector-valueddistributions.

54

TheSpacesLp,µandWeightedAnisotropicSpaces

Lemma1.3.18.LetEbeaBanachspaceofclassHT,σ>0,p,q∈(1,∞),andlet
hk∈Lp(Rn;E),k∈N0,satisfy
suppFhk⊂{|ξ|≤2k+3},k≥0.
If(2kσ|hk|Lp(Rn;E))k∈N0∈lqthenk∞=0hkconvergestosomeh∈Bσp,q(Rn;E)inthesense
ofdistributions,anditholds
|h|Bσp,q(Rn;E)|(2kσ|hk|Lp(Rn;E))k∈N0|lq.
Proof.ThesupportconditionimpliesthatSjhk=0forj≥k+4.Thus
NNSjhk=Sjhk1[0,∞)(k),j,N∈N0.
k=0k=j−3
(I)WefirstshowthatkN=0hkN∈NisconvergentinthesenseofdistributionsasN→∞.
ForintegerN1<N2itholds,usingtheuniformboundednessof(Sj)j∈N0,
N2jσ/2N2
|hk|Bp,σ/∞2(Rn;E)=j∈supN02|Sjhk|Lp(Rn;E)
k=N1k=N1
N2sup2(j/2−k)σ2kσ|hk|Lp(Rn;E)
j∈N0k=max{N1,j−3}
N2sup2(j/2−k)σ,
j∈N0k=max{N1,j−3}
kN=0hkN∈NisaCauchysequenceinBp,σ/∞2(Rn;E),andthusconvergesinthesense
whichissmallerthananygivennumberifN1<N2aresufficientlylarge.Therefore
ofdistributionstoafunctionh.
(II)Weshowthatinfacth∈Bσp,q(Rn;E).TothisendweestimateforN∈N
NN|hk|Bσp,q(Rn;E)=|(2jσ|Sjhk1[0,∞)(k)|Lp(Rn;E))j∈N0|lq
k=0k=j−3
N|(2jσ|hk1[0,∞)(k)|Lp(Rn;E))j∈N0|lq
∞k=j−3
≤|(2jσ|hj+l1[0,∞)(j+l)|Lp(Rn;E))j∈N0|lq
3−=l∞≤2−lσ|(2kσ|hk|Lp(Rn;E))k∈N0|lq|(2kσ|hk|Lp(Rn;E))k∈N0|lq,
3−=lwhichyieldsthatkN=0hkN∈NisuniformlyboundedinBσp,q(Rn;E).SinceEisassumed
tobereflexivethesequencehasaweaklyconvergentsubsequenceinBσp,q(Rn;E),andin
particular,thisconvergenceisinthedistributionalsense.Fromtheuniquenessofdistribu-
tionallimitsweobtainh∈Bσp,q(Rn;E).
AfterthesepreparationswecanestimatetheBesovnormofaproductinawaythatis
suitableforourpurposes.

eightedW1.3SpacesAnisotropic

55

Lemma1.3.19.LetEbeaBanachspaceofclassHT,andletΩ⊂Rnbeadomainwith
compactsmoothboundary,orΩ∈{Rn,R+n}.Letfurtherσ>0,p,q∈(1,∞),andassume
thatr,r,ρ,ρ∈[p,∞],r,ρ=∞,satisfyr1+r1=ρ1+ρ1=p1.Thenitholds
|fg|Bσp,q(Ω;E)|f|Lρ(Ω;B(E))|g|Bρσ,q(Ω;E)+|f|Brσ,q(Ω;B(E))|g|Lr(Ω;E).
InthisestimateΩ=Rnmaybereplacedbyitsboundary∂Ω.
Weremarkthatofparticularinterestisherethecaseρ=r=∞.
Proof.(I)WefirstconsiderthecaseΩ=Rn,andestimatetheparaproductsforfggiven
by(1.3.25).UsingLemma1.3.18,Hölder’sinequalityinLp(Rn;E)andLemma1.3.17we
estimateforΠ1(f,g)
∞|Sk−2fSkg|Bσp,q(Rn;E)|(2kσ|Sk−2fSkg|Lp(Rn;E))k≥2|lq
=2k≤|(2kσ|Sk−2f|Lρ(Rn;B(E))|Skg|Lρ(Rn;E))k≥2|lq
≤j∈supN0|Sjf|Lρ(Rn;B(E))|(2kσ|Skg|Lρ(Rn;E))k≥2|lq
|f|Lρ(Rn;B(E))|g|Bρσ,q(Rn;E).
InasimilarwayweobtainforΠ2(f,g),withl∈{−1,0,1},
∞|Sk+lfSkg|Bσp,q(Rn;E)|(2kσ|Sk+lf|Lr(Rn;B(E))|Skg|Lr(Rn;E))k∈N0|lq
=0kσk≤|(2|Sk+lf|Lr(Rn;B(E)))k∈N0|lqj∈supN0|Sjg|Lr(Rn;E)
|f|Brσ,q(Rn;B(E))|g|Lr(Rn;E),

andforΠ3(f,g)
∞|SkfSk−2g|Bσp,q(Rn;E)|(2kσ|Skf|Lr(Rn;B(E))|Sk−2g|Lr(Rn;E))k≥2|lq
=2k|f|Brσ,q(Rn;B(E))|g|Lr(Rn;E).
Thustheparaproductsexistinthesenseofdistributions,withthegivenestimates.This
yieldstheassertionforΩ=Rn.
(II)TheestimateforgeneralΩmaybeobtainedfromthefull-spacecaseusingtheex-
tensionoperatorEΩfrom(1.3.2).ItislefttoshowtheestimateforΩ=Rnreplacedby
itsboundary.Wedescribe∂Ωbyafinitecollectionofcharts(Ui,ϕi)andapartitionof
unity{ψi}subordinatetothecoveriUi.ForeachiwechooseanopensetWi⊂Rnwith

56

TheSpacesLp,µandWeightedAnisotropicSpaces

suppψi⊂Wi⊂Wi⊂Uiandestimate,usingLemmaA.4.1,
1−1−|fg|Bσp,q(∂Ω;E)i|f(ϕi)g(ϕi)|Bσp,q(ϕi(Wi)∩Rn−1;E)
1−1−i|f(ϕi)|Lr(ϕi(Wi)∩Rn−1;B(E))|g(ϕi)|Brσ,q(ϕi(Wi)∩Rn−1;E)
1−1−+|f(ϕi)|Brσ,q(ϕi(Wi)∩Rn−1;B(E))|g(ϕi)|Lr(ϕi(Wi)∩Rn−1;E)
1−i|f|Lr(∂Ω;B(E))|g(ϕi)|Brσ,q(ϕi(Wi)∩Rn−1;E)
1−+|f(ϕi)|Brσ,q(ϕi(Wi)∩Rn−1;B(E))|g|Lr(∂Ω;E).
Nowtakeafunctionψi∗∈Cc∞(Ui)withψi∗≡1onWi,todeduceagainfromLemmaA.4.1
that|fg|Bσp,q(∂Ω;E)|f|Lr(∂Ω;B(E))|ψi∗(ϕi−1)g(ϕi−1)|Bσ(ϕi(Ui)∩Rn−1;E)
,qri1−1−∗+|ψi(ϕi)f(ϕi)|Brσ,q(ϕi(Ui)∩Rn−1;B(E))|g|Lr(∂Ω;E)
|f|Lr(∂Ω;B(E))|g|Brσ,q(∂Ω;E)+|f|Brσ,q(∂Ω;B(E))|g|Lr(∂Ω;E).
Wederiveasimilarresultforcertainvector-valuedBesovspacesonthehalf-line.
Lemma1.3.20.LetEbeaBanachspaceofclassHT,andletΩ⊂Rnbeadomainwith
compactsmoothboundary,orΩ∈{Rn,R+n}.Letfurtherσ>0,p,q∈(1,∞),andassume
thatr,r,ρ,ρ,s,s,σ,σ∈[p,∞],s,σ=∞,satisfyr1+r1=ρ1+ρ1=s1+s1=σ1+σ1=p1.
holdsitThen|fg|Bσp,q(R+;Lp(Ω;E))|f|Lσ(R+;Lρ(Ω;B(E)))|g|Bσσ,q(R+;Lρ(Ω;E))
+|f|Bσs,q(R+;Lr(Ω;B(E)))|g|Ls(R+;Lr(Ω;E)).
InthisestimateΩ=Rnmaybereplacedbyitsboundary∂Ω.
Proof.Usingextensionsandrestrictions,wemayconsidertheestimateonRinsteadof
R+.Weproceedasinthepreviouslemma.ForΠ1(f,g)weobtain,usingHölder’sinequality
wice,t

∞|Sk−2fSkg|Bσp,q(R;Lp(Ω;E))|(2kσ|Sk−2fSkg|Lp(R;Lp(Ω;E)))k≥2|lq
=2k≤|(2kσ|Sk−2f|Ls(R;Lr(Ω;B(E)))|Skg|Ls(R;Lr(Ω;E)))k≥2|lq
≤sup|Sjf|Ls(R;Lr(Ω;B(E)))|(2kσ|Skg|Ls(R;Lr(Ω;E)))k≥2|lq
N∈j0|f|Ls(R;Lr(Ω;B(E)))|g|Bsσ,q(R;Lr(Ω;E)).
InasimilarwayonetreatsthetermsΠ2(f,g)andΠ3(f,g).AsintheproofofLemma1.3.19
thisimpliestheassertedestimate.SinceforthespatialvariablesonlyHölder’sinequality
wasused,onemayreplaceΩ=Rnbyitsboundaryintheabovearguments.
Wecannowprovethedesiredsufficientconditionsforpointwisemultipliersonboundaries.
WestartwithspatialBesovregularity.

eightedW1.3SpacesAnisotropic

57

Lemma1.3.21.LetEbeaBanachspaceofclassHT,p∈(1,∞),µ∈(1/p,1],let
J=(0,T)befinite,andletΩ⊂Rnbeadomainwithcompactsmoothboundary∂Ω,or
Ω=R+n.Letfurthers,r∈[p,∞),m∈N,κ∈(0,1),τ∈(0,2)andϑ>0satisfy
τ>κ,ϑ>2mκ,p(1−µ+1/p)<1−n−1τ.
ϑrsholdsitThen|bu|Lp,µ(J;B2p,pmκ(∂Ω;E))|b|Ls(J;Br2,pmκ(∂Ω;B(E))|u|Wτp,µ(J;Lp(∂Ω;E))∩Lp,µ(J;Bϑp,p(∂Ω;E)).(1.3.26)
Moreover,forp(1−µs+1/p)+2n−mr1<κthereisδ∈(0,κ)suchthat
|bu|Lp,µ(J;B2p,pmκ(∂Ω;E))|b|L∞(J;L∞(∂Ω;B(E)))|u|Wκp,µ(J;Lp(∂Ω;E))∩Lp,µ(J;B2p,pmκ(∂Ω;E))(1.3.27)
+|b|Ls(J;Br2,pmκ(∂Ω;B(E))|u|Wκp,µ−δ(J;Lp(∂Ω;E))∩Lp,µ(J;B2p,pm(κ−δ)(∂Ω;E)).
Restrictingtou∈0Wp,µ,forgivenT0>0theseestimatesholdwithauniformconstant
forallT≤T0.
Proof.(I)Itholdsthat
pp|bu|Lp,µ(J;B2p,pmκ(∂Ω;E))=Jtp(1−µ)|b(t,∙)u(t,∙)|B2p,pmκ(∂Ω;E))dt,
andforalmosteveryt∈JweuseLemma1.3.19toestimate
|b(t,∙)u(t,∙)|B2p,pmκ(∂Ω;E))|b(t,∙)|Lρ(∂Ω;B(E))|u(t,∙)|Bρ2,pmκ(∂Ω;E)
+|b(t,∙)|Br2,pmκ(∂Ω;B(E))|u(t,∙)|Lr(∂Ω;E),
wherer1+r1=ρ1+ρ1=p1.Hölder’sinequalitynowyields
|bu|Lp,µ(J,B2p,pmκ(∂Ω;E))|b|Lσ(J;Lρ(∂Ω;B(E)))|u|Lσ,µ(J;Bρ2,pmκ(∂Ω;E))
+|b|Ls(J;Br2,pmκ(∂Ω;B(E)))|u|Ls,µ(J;Lr(∂Ω;E)),(1.3.28)
wheres1+s1=σ1+σ1=p1.Toobtainthedesiredestimateswehavetochoosethese
numbersappropriately.
(II)Westartwiththefirstsummandin(1.3.28).Ifτ=κwetakeρ=σ=∞,ρ=σ=p,
andobtainthefirstsummandontheright-handsideof(1.3.27).
Nowsupposethatτ>κ.Thenwetakeσ=sandσ=s.Theembedding
Ls(J;Br2,pmκ(∂Ω;B(E)))→Ls(J;Lρ(∂Ω;B(E)))
isvalidfor112mκ112mκ
ρ>r−n−1,i.e.,ρ<r+n−1.(1.3.29)
Wethusneedtheembedding
Wτp,µ(J;Lp(Γ;E))∩Lp,µ(J;Bϑp,p(Γ;E))→Ls,µ(J;Bρ2,pmκ(∂Ω;E))(1.3.30)

58

TheSpacesLp,µandWeightedAnisotropicSpaces

forsomeρthatsatisfies(1.3.29).DuetoProposition1.3.2,forgivenθ∈(0,1)itholds
Wτp,µ(J;Lp(∂Ω;E))∩Lp,µ(J;Bϑp,p(∂Ω;E))→Wp,µ(1−θ)τ(J;Hpϑθ(∂Ω;E)),
whereheretheembeddingconstantisindependentofJifonerestrictsto0Wp,µ,andfurther
theSobolevembedding
Hpϑθ(∂Ω;E)→B2ρmκ,p(∂Ω;E)forθ>2mκ+n−1(1.3.31)
ϑρϑisvalid.Therefore,ifwechooseθ>nϑr−1,then(1.3.31)holdswithsomeρthatsatisfies
(1.3.29).FromProposition1.1.12weinfer
Wp,µ(1−θ)τ(J;Bρ2,pmκ(∂Ω;E))→Ls,µ(J;Bρ2,pmκ(∂Ω;E))
for(1−θ)τ−(1−µ+p1)>−p(1−sµ+1/p),withauniformembeddingconstantinthe
0Wp,µ-case.Sinceitisassumedthatp(1−µs+1/p)<1−nϑr−1τ,theaboveinequalityholds
forallθ>nϑr−1whicharesufficientlyclosetonϑr−1.Thisyields(1.3.30)withsomeρthat
satisfies(1.3.29),andweobtainanestimateofthefirstsummandin(1.3.28)appropriate
(1.3.26).for(III)Toestimatethesecondsummandin(1.3.28)wehavetoshowthat
Wτp,µ(J;Lp(∂Ω;E))∩Lp,µ(J;Bϑp,p(∂Ω;E))→Ls,µ(J;Lr(∂Ω;E)),
withτreplacedbyκ−δandϑreplacedby2m(κ−δ)for(1.3.27),whereδ∈(0,κ).
UsingProposition1.3.2,itcanbeseenasabovethatthisembeddingholdsifp(1−µs+1/p)<
1−nϑr−1τ,withtherespectivereplacementsfor(1.3.27)andthedependenceonJinthe
0Wp,µ-caseasasserted.Thisyields(1.3.26)and(1.3.27),respectively.
WenextconsidertemporallyweightedSlobodetskiiregularity.
Lemma1.3.22.LetEbeaBanachspaceofclassHT,p∈(1,∞),µ∈(1/p,1],let
J=(0,T)befinite,andletΩ⊂Rnbeadomainwithcompactsmoothboundary∂Ω,or
Ω=R+n.Letfurthers,r∈[p,∞),m∈N,κ∈(0,1),κ=1−µ+1/p,τ∈(0,2)andϑ>0
satisfyτ>κ,ϑ>2mκ,p(1−µ+1/p)<1−n−1τ,
ϑrsthatfurtherosesuppand1−1−µ+1/p>n−1ifκ>1−µ+1/p.(1.3.32)
ϑrτholdsitThen|bu|Wκp,µ(J;Lp(∂Ω;E))|b|Bκs,p(J;Lr(∂Ω;B(E)))|u|Wτp,µ(J;Lp(∂Ω;E))∩Lp,µ(J;Bϑp,p(∂Ω;E)).(1.3.33)
Moreover,ifp(1−µs+1/p)+2n−mr1<κand
κ−(1−µ+1/p)∈/0,n−1,(1.3.34)
mr2

(1.3.34)

eightedW1.3SpacesAnisotropic

59

thenthereisδ∈(0,κ)suchthat
|bu|Wκp,µ(J;Lp(∂Ω;E))|b|L∞(J;L∞(∂Ω;B(E)))|u|Wκp,µ(J;Lp(∂Ω;E))∩Lp,µ(J;B2p,pmκ(∂Ω;E))(1.3.35)
+|b|Bκs,p(J;Lr(∂Ω;B(E)))|u|Wκp,µ−δ(J;Lp(∂Ω;E))∩Lp,µ(J;B2p,pm(κ−δ)(∂Ω;E)).
Restrictingtou∈0Wp,µ,andassumingthatbisdefinedonalargerintervalJ0=(0,T0),
T0>0,theseestimates,withJreplacedbyJ0inthenormsforb,holdwithauniform
constantforallT≤T0.Inthiscase,andfurtherintheunweightedcaseµ=1,onecan
(1.3.34).and(1.3.32)ignoreProof.(I)WeintendtouseLemma1.3.20,andthushavetoreducetheestimatetothe
unweightedcaseonR+.Ifκ>1−µ+1/pthenitisassumedthat1−1−µτ+1/p>nϑr−1.In
thiscaseitfollowsfromthePropositions1.3.2and1.1.11that
Wτp,µ(J;Lp(∂Ω;E))∩Lp,µ(J;Bϑp,p(∂Ω;E))→BUC(J;Lr(∂Ω;E)),(1.3.36)
suchthatu0:=u(0,∙)∈Lr(∂Ω;E)iswelldefined.Moreover,ifκ<1−µ+1/pandin
the0Wp,µ-casewesetu0:=0.Inbothcaseswehave
|bu|Wκp,µ(J;Lp(∂Ω;E))≤|b(u−u0)|Wκp,µ(J;Lp(∂Ω;E))+|bu0|Wκp,µ(J;Lp(∂Ω;E)).(1.3.37)
Incaseκ>1−µ+1/pweuseinterpolationand(1.3.36)toobtainthatforthesecond
summandin(1.3.37)itholds
|bu0|Wκp,µ(J;Lp(∂Ω;E))|b|Bκp,p(J;Lr(∂Ω;B(E)))|u0|Lr(∂Ω;E)
|b|Bκs,p(J;Lr(∂Ω;B(E)))|u|Wτp,µ(J;Lp(∂Ω;E))∩Lp,µ(J;Bϑp,p(∂Ω;E)),(1.3.38)
asdesiredfor(1.3.33).Replacingτbyκandϑby2mκ,andnotingthattheconditiononκ
isstrict,(1.3.38)yieldsatermasinthesecondsummandof(1.3.35),withsomeδ∈(0,κ).
(II)Toestimatethefirstsummandin(1.3.37)wesetv:=u−u0,suchthat
v∈0Wκp,µ(J;Lp(∂Ω;E))
inanycase,duetotheassumptionκ=1−µ+1/pandProposition1.1.11.Itfurther
followsfromProposition1.1.11,Lemma1.1.3,withthehelpoftheextensionoperatorsEJ
andEJ0fromLemma1.1.5,andfromLemma1.3.20,that
|bv|Wκp,µ(J;Lp(∂Ω;E))|bv|0Wκp,µ(J;Lp(∂Ω;E))|b(t1−µv)|0Wpκ(J;Lp(∂Ω;E))
≤|(EJb)(EJ0t1−µv)|0Wpκ(R+;Lp(∂Ω;E))|(EJb)(EJ0t1−µv)|Wpκ(R+;Lp(∂Ω;E))
|EJb|Lσ(R+;Lρ(∂Ω;B(E)))|(EJ0t1−µv)|Bσκ,p(R+;Lρ(∂Ω;E))
+|EJb|Bκs,p(R+;Lr(∂Ω;B(E)))|(EJ0t1−µv)|Ls(R+;Lr(∂Ω;E))
|EJb|Lσ(R+;Lρ(∂Ω,B(E)))|t1−µv|0Bσκ,p(J;Lρ(∂Ω;E))
+|EJb|Bκs,p(R+;Lr(∂Ω;B(E)))|v|Ls,µ(J;Lr(∂Ω;E)),(1.3.39)
wherer1+r1=ρ1+ρ1=p1ands1+s1=σ1+σ1=p1,sothats,σ≥p,havetobechosen
appropriately.Inthe0Wp,µ-case,assumingthatbisdefinedonJ0=(0,T0)withT≤T0,

60

TheSpacesLp,µandWeightedAnisotropicSpaces

in(1.3.39)wecanreplaceJbyJ0andEJbyEJ0,whichleadstoconstantsin(1.3.33)and
desired.as(1.3.35)(III)Weconsiderthefirstsummandin(1.3.39).Forτ=κwetakeρ=σ=∞,ρ=σ=p,
anddeducefromLemma1.1.3and(1.3.36)that
|t1−µv|0Wpκ(J;Lp(∂Ω;E))|u|Wκp,µ(J;Lp(∂Ω;E))+|u0|Wκp,µ(J;Lp(∂Ω;E))|u|Wκp,µ(J;Lp(∂Ω;E)),
wheretheconstantforthisestimateisofcourseindependentofJifu0=0.Thisyields
thefirstsummandin(1.3.35)asdesired.Further,forτ>κwetakeρ=r,ρ=r,and
estimateforsufficientlysmallε>0
µ−1|tv|0Bσκ,p(J;Lr(∂Ω;E))|v|Wσκ+,µε(J;Lr(∂Ω;E))
≤|u|Wσκ+,µε(J;Lr(∂Ω;E))+|u0|Wσκ+,µε(J;Lr(∂Ω;E))
|u|Wσκ+,µε(J;Lr(∂Ω;E))+|u|Wτp,µ(J;Lp(∂Ω;E))∩Lp,µ(J;Bϑp,p(∂Ω;E)),
withauniformconstantinthe0Wp,µ-case.Observethat
Bκs,p(R+;Lr(∂Ω;B(E)))→Lσ(R+;Lr(∂Ω;B(E)))if1<1+κ.(1.3.40)
sσSincealloccurringrelationsarestrict,itthussufficestoshowthatthereexistsaσ≥p,
thathsuc(1.3.40),satisfyingWτp,µ(J;Lp(∂Ω;E))∩Lp,µ(J;B2p,pmτ(∂Ω;E))→Wσκ,µ(J;Lr(∂Ω;E))
holdstrue.Forθ∈(0,1),Proposition1.3.2yields
Wτp,µ(J;Lp(∂Ω;E))∩Lp,µ(J;Bϑp,p(∂Ω;E))→W(1p,µ−θ)τ(J;Hpϑθ(∂Ω;E)),
withauniformembeddingconstantinthe0Wp,µ-case.Itholds
Hpϑθ(∂Ω;E)→Lr(∂Ω;E)ifθ>n−1,(1.3.41)
ϑrandmoreoveritfollowsfromProposition1.1.12that
Wp,µ(1−θ)τ(J;Lr(∂Ω;E))→Wσκ,µ(J;Lr(∂Ω;E))
if(1−θ)τ−(1−µ+1/p)>κ−p(1−µ+1/p),(1.3.42)
σagainwithauniformconstantinthe0Wp,µ-case.Wecannowchooseθandσ≥psatisfying
(1.3.40)and(1.3.36)suchthat(1.3.42)holds,usingp(1−µs+1/p)<1−nϑr−1τ.Sowehave
showntheassertedestimatesforthefirstsummandin(1.3.39).
(IV)Forthesecondsummandin(1.3.39)itholds,asabove,
|v|Ls,µ(J;Lr(∂Ω;E))|u|Ls,µ(J;Lr(∂Ω;E))+|u|Wτp,µ(J;Lp(∂Ω;E))∩Lp,µ(J;Bϑp,p(∂Ω;E)).
UsingthePropositions1.3.2and1.1.12,itcanbeseenasinthepreviousstepthat
Wτp,µ(J;Lp(∂Ω;E))∩Lp,µ(J;Bϑp,p(∂Ω;E))→Ls,µ(J;Lr(∂Ω;E))

(1.3.41)

1.3WeightedAnisotropicSpaces61
isvalidifp(1−µs+1/p)<1−nϑr−1τ,withauniformconstantinthe0Wp,µ-case.Thisshows
(1.3.33).For(1.3.35)thesameargumentsarevalidwithτreplacedbyκ−δandϑreplaced
by2m(κ−δ)withδ∈(0,κ).
Itseemsthattheexceptions(1.3.32)and(1.3.34)arenotessentialandonlyduetoour
proof.IfonehadaLittlewood-PaleyrepresentationofthespacesWsp,µ,asfortheun-
weightedSlobodetskiispaces,thenonecouldargueasinLemma1.3.20andalsocoverthe
alues.vexceptional

Theaboveresultsonpointwisemultiplicationarerathersharp,andvalueableforlowvalues
ofp∈(1,∞),comparedtoκ.Itturnsoutthatifpissufficientlylargethenthespaces
Wp,µκ,2mκ(J×∂Ω)13areclosedunderpointwisemultiplications,andthenalsobmaybelong
toatemporallyweightedspace.
Lemma1.3.23.LetEbeofclassHT,letJ=(0,T)befiniteorinfiniteandletp∈(1,∞),
µ∈(1/p,1]andκ∈(0,1),ϑ∈R+\N.Thenitholds
|bu|Wp,µκ,ϑ(J×∂Ω;E)|b|L∞(J×∂Ω,B(E))|u|Wp,µκ,ϑ(J×∂Ω;E)+|b|Wp,µκ,ϑ(J×∂Ω;B(E))|u|L∞(J×∂Ω,E).
Moreover,if
1−1−µ+1/pϑ>n−1,(1.3.43)
pκestimateanistherethen|bu|Wp,µκ,ϑ(J×∂Ω;E)|b|Wp,µκ,ϑ(J×∂Ω;B(E))|u|Wp,µκ,ϑ(J×∂Ω;E).
ReplacingWp,µκ,ϑby0Wp,µκ,ϑ,theseestimatesareindependentofthelengthofJ.Ifbisdefined
onalargerintervalJ0=(0,T0),T≤T0,andonerestrictstoa0Wp,µκ,ϑ-spaceforu,then
theestimatesareuniforminT.
Proof.Throughoutwedenoteanyoccurringsup-normby|∙|∞.
(I)ByLemma1.3.19wehaveforalmostallt∈Jthat
|b(t,∙)u(t,∙)|Wpϑ(∂Ω;E)|b(t,∙)|∞|u(t,∙)|Wpϑ(∂Ω;E)+|b(t,∙)|Wpϑ(∂Ω;B(E))|u(t,∙)|∞,
andweobtaintheassertedestimatefor|bu|Lp,µ(J;Wpϑ(∂Ω;E))bytakingtheLp,µ-norm.This
estimateisalwaysindependentofthelengthofJ.ForWκp,µ(J;Lp(∂Ω;E))weusethe
intrinsicnormgivenbyProposition1.1.13toobtain
|bu|Wκp,µ(J;Lp(∂Ω;E))|b|∞|u|Lp,µ(J;Lp(∂Ω;E))(1.3.44)
+|b|∞[u]Wκp,µ(J;Lp(∂Ω;E))+[b]Wκp,µ(J;Lp(∂Ω;B(E)))|u|∞.
NotethatthisestimatealsoholdstrueforJ=R+.
(II)Nowletb,u∈0Wp,µκ,ϑ.TogetanestimateindependentofJwecannotusetheintrin-
sicnormfor0Wκp,µdirectly(seethediscussioninRemark1.1.15).Wethereforetakethe
13RecallthenotationWp,µκ,ϑ(J×∂Ω;E)=Wκp,µ(J;Lp(∂Ω;E))∩Lp,µ(J;Wpϑ(∂Ω;E)).

62

TheSpacesLp,µandWeightedAnisotropicSpaces

extensionoperatorEJ0fromProposition1.1.5,whosenormisindependentofthelengthof
J,andestimate,usingProposition1.1.11and(1.3.44)onthehalf-line,
|bu|0Wκp,µ(J;Lp(∂Ω;E))≤|EJ0bEJ0u|Wκp,µ(R+;Lp(∂Ω;E))
≤|EJ0b|∞|EJ0u|Wκp,µ(R+;Lp(∂Ω;E))+|EJ0b|Wκp,µ(R+;L(∂Ω;B(E)))|EJ0u|∞
|b|∞|u|0Wκp,µ(J;Lp(∂Ω;E))+|b|0Wκp,µ(J;Lp(∂Ω;B(E)))|u|∞.
TheseestimatesareindependentofthelengthofJ.Ifbisdefinedonalargerinterval
J0=(0,T0),thenonemayreplaceEJ0byEJ0intheaboveargumentstoobtainanestimate
uniformlyinTinthiscase.
(III)Finally,if(1.3.43)isvalidthentheassertedestimatefollowsfrom
Wp,µκ,ϑ(J×∂Ω;E)→C(J×∂Ω;E),(1.3.45)
whichisduetoProposition1.3.2andSobolev’sembeddings,andisindependentofthe
lengthofJinthe0Wp,µκ,ϑ-caseandindependentofT≤T0ifbisdefinedonJ0.
Weemphasizethatforϑ=2mκthecondition(1.3.43)isequivalenttoκ>1−µ+1/p+2n−mp1.
Wesummarizetheabovepointwisemultiplicationresultsforthecoefficientsofboundary
differentialoperatorsasfollows.
Proposition1.3.24.LetEbeaBanachspaceofclassHT,p∈(1,∞),µ∈(1/p,1],let
J=(0,T)beafiniteinterval,andletΩ⊂Rnbeadomainwithcompactsmoothboundary
∂Ω,orΩ=R+n.Letfurtherm∈Nandk∈N0,k≤2m−1,definethenumber
1kκ:=1−2m−2mpandsupposethatκ=1−µ+1/p.
AssumethatfortheB(E)-valuedcoefficientb=b(t,x)oftheoperatorbtrΩβ,where
β∈N0nwith|β|≤k,oneofthefollowingtwoconditionsisvalid:either
κ>1−µ+1/p+n−1andb∈Wp,µκ,2mκ(J×∂Ω;B(E)),(1.3.46)
mp2holdsitorb∈Bsκβ,p(J;Lrβ(∂Ω;B(E)))∩Lsβ(J;Br2βmκ,p(∂Ω;B(E))),(1.3.47)
withnumberssβ,rβ∈[p,∞)sothat
p(1−µ)+1+n−1<κ+k−|β|,κ+k−|β|−(1−µ+1/p)∈/0,n−1.
sβ2mrβ2m2m2mrβ
Theninbothcaseswehave
btrΩβ∈BW1p,µ,2m(J×Ω;E),Wp,µκ,2mκ(J×∂Ω;E),
andif|β|=kthen
b∈BUC(J×∂Ω;B(E)),|β|=mj,j=1,...,m.

SpacesAnisotropiceightedW1.3

63

Proof.ItfollowsfromLemma1.3.4andProposition1.3.12thattrΩβmaps
W1p,µ,2m(J×Ω;E)→W1p,µ−|β|/2m−1/2mp,2m−|β|−1/p(J×∂Ω;E)
inacontinuousway.ObservethatthelatterspaceembedsintoWp,µκ,2mκ(J×∂Ω;E),since
|β|≤k.Assumethat(1.3.46)holds.ThenLemma1.3.23impliesthatbtrΩβmapscon-
tinuouslyasasserted.Thecontinuityofbfollowsfrom(1.3.45).
Nextassume(1.3.47).ThenwecanapplytheLemmas1.3.21and1.3.22withτ=1−
|β|/2m−1/2mpandϑ=2mτtoobtaintheassertedmappingpropertyofbtrΩβ.Incase
|β|=kitholds
Bsκβ,p(J;Lrβ(∂Ω;B(E)))∩Lsβ(J;Br2βmκ,p(∂Ω;B(E)))→C(J×∂Ω;B(E)),
whichfollowsfromProposition1.3.2,theremarkthereafter,Proposition1.1.11and
Sobolev’sembeddings,andshowsthatbisacontinuousfunction.
Wefinishthissectionwithatechnicalresultoncompatibledataontheboundary.
Lemma1.3.25.InthesituationofProposition1.3.24,forκ>1−µ+1/pthesets
D:=(g,u0)∈Wp,µκ,2mκ(J×∂Ω;E)×B2p,pm(µ−1/p)(Ω;E):b(0,∙)trΩβu0=g(0,∙)onΓ,
D0:=(g,u0)∈D:g∈0Wp,µκ,2mκ(J×∂Ω;E),
arewell-definedandclosedsubspacesofWp,µκ,2mκ(J×∂Ω;E)×B2p,pm(µ−1/p)(Ω;E)and
0Wp,µκ,2mκ(J×∂Ω;E)×B2p,pm(µ−1/p)(Ω;E),respectively.
Proof.If(1.3.46)isvalid,thenb(0,∙)alwaysexists.Toobtainthisincase(1.3.47),note
thatκ>1−µ+1/pinparticularyieldsκ>1/s.HenceDandD0arewell-definedinboth
cases.ToshowthatDisclosedinWp,µκ,2mκ(J×∂Ω;E)×B2p,pm(µ−1/p)(Ω;E)takeasequence
(gk,u0k)k∈N∈D,andassumethat(gk,u0k)convergesto(g,u0)ask→∞withrespect
tothenormofWp,µκ,2mκ(J×∂Ω;E)×B2p,pm(µ−1/p)(Ω;E).Itisthenaconsequenceof(1.3.20)
andTheorem1.3.6that(gk(0,x),trΩβu0k(x))k∈Nconverges(uptoasubsequence)to
(g(0,x),trΩβu0(x))ask→∞,foralmosteveryx∈∂Ω.Moreover,forallktheidentity
b(0,x)trΩβu0k(x)=gk(0,x)
isvalidforalmosteveryx∈∂Ω.Takingthelimit,weobtainthatb(0,x)trΩβu0(x)=
g(0,x)holdstrueforallx∈∂Ωwhicharenotcontainedinacountableunionofsubsets
ofsurfacemeasurezeroof∂Ω.Thisyields(g,u0)∈D.TheclosednessofD0followsfrom
ts.argumensamethe

Chapter2

MaximalLp,µ-RegularityforStatic
ConditionsBoundary

InthischapterwedevelopthemaximalLp,µ-regularityapproachforageneralclassof
parabolicinitial-boundaryvalueproblemswithinhomogeneousstaticboundarycondi-
tions,generalizingtheresultsofDenk,Hieber&Prüss[25].InSection2.1wedescribe
theapproachandtheinvolvedfunctionspacesindetail,provideexamples,describethe
advantagescomparedtotheunweightedapproach,andgiveanoutlineofthestrategyhow
toobtainthemainresultofthepresentchapter,Theorem2.1.4.Theproofofthetheorem
iscarriedoutindetailintheSections2.2,2.3,and2.4,andfollows[25].InSection2.5we
showthatrelatedboundaryoperatorsadmitacontinuousright-inverse.

2.1TheProblemandtheApproachinWeightedSpaces

ProblemTheFortheunknownu=u(t,x)∈Eweconsiderthelinearinhomogeneous,nonautonomous
parabolicinitial-boundaryvalueproblem

∂tu+A(t,x,D)u=f(t,x),x∈Ω,t∈J,
Bj(t,x,D)u=gj(t,x),x∈Γ,t∈J,j=1,...,m,(2.1.1)
u(0,x)=u0(x),x∈Ω.
WeassumethatΩ⊂RnisadomainwithcompactsmoothboundaryΓ=∂Ω,that
J=(0,T)isafiniteinterval,T>0,andthatEisacomplexBanachspaceofclassHT.
ThedifferentialoperatorAoforder2m,wherem∈N,isgivenby
A(t,x,D)=aα(t,x)Dα,x∈Ω,t∈J,
m2|≤α|whereD=−i,and=(∂x1,...,∂xn)denotestheeuclidiangradientonRn.Thedynamic
equationinthedomainiscomplementedbymboundaryconditionsoforderatmost2m−1.

66

MaximalLp,µ-RegularityforStaticBoundaryConditions

TheboundaryoperatorsBjareoftheform
Bj(t,x,D)=bjβ(t,x)trΩDβ,x∈Γ,t∈J,j=1,...,m,
m|≤β|jwheretrΩdenotesthetraceonΩ,andwheretheintegermj∈{0,...,2m−1}istheorderof
Bj.ObservehowBjactsonafunctionu:onefirstappliesthecomponentsoftheeuclidian
gradientandthenthespatialtrace.WeassumethateachoftheoperatorsBjisnontrivial,
Bj=0,andwrite
B:=(B1,...,Bm).
ThecoefficientsoftheoperatorstakevaluesintheboundedlinearoperatorsonE,i.e.,
aα(t,x)∈B(E),x∈Ω,t∈J,|α|≤2m,
bjβ∈B(E),x∈Γ,t∈J,|β|≤mj,j=1,...,m.
Finally,thedataontheright-handsideisE-valued,andisassumedtobegiven.
Example2.1.1.Weconsidertwoproblemsthatfitintotheaboveframework.Thefirst
isalinearizedreaction-diffusionsystem,givenby
∂tu−Δu=f(t,x),x∈Ω,t∈J,
∂νu=g(t,x),x∈Γ,t∈J,
u(0,x)=u0(x),x∈Ω,
where∂ν=ν∙trΩdenotesthederivativewithrespecttotheouterunitnormalfieldνof
Γ.Here,theorderofA(D)=−Δis2,thuswehavem=1,andtheorderoftheboundary
operatorB1(x,D)=∂νism1=1.
AfurtherproblemthatfitsintoourframeworkisalinearizedCahn-Hilliardphasefield
model,givenby

∂tu+Δ2u−Δu=f(t,x),x∈Ω,t∈J,
−∂νΔu+∂νu=g1(t,x),x∈Γ,t∈J,
∂νu=g2(t,x),x∈Γ,t∈J,
u(0,x)=u0(x),x∈Ω.
HereA(D)=Δ2−Δisoforder4,whichmeansm=2,andthedynamicequationinΩ
iscomplementedbytwoboundaryconditions,withB1(x,D)=−∂νΔ+∂ν,m1=3,and
B2(x,D)=∂ν,m2=1.

TheApproachintheLp,µ-spaces
WedescribethemaximalLp,µ-regularityapproachfor(2.1.1).Let
p∈(1,∞),µ∈(1/p,1].

2.1TheProblemandtheApproachinWeightedSpaces

67

Thebasisoftheapproachisthatthedomaininhomogeneityfandthesolutionushall
satisfyf,u,∂tu,Au∈E0,µ:=Lp,µ(J;Lp(Ω;E)).
Theseassumptionsdeterminetheregularityofuandtheotherdataasfollows.SinceAis
oforder2m,itshouldhold
u∈Eu,µ:=W1p,µ(J;Lp(Ω;E))∩Lp,µ(J;Wp2m(Ω;E))
forthesolutionof(2.1.1).Fortheinitialvalue,Theorem1.3.6ontemporaltracesyields
u0∈Xu,µ:=B2p,pm(µ−1/p)(Ω;E).
Fortheboundaryinhomogeneities,sincetheoperatorBjisofordermj,asuccessiveap-
plicationofLemma1.3.4onspatialderivatives,togetherwithProposition1.3.12onthe
spatialtraceonanisotropicspacesyields
gj∈Fj,µ:=Wκp,µj(J;Lp(Γ;E))∩Lp,µ(J;Wp2mκj(Γ;E)),j=1,...,m,
wherethenumberκj∈(0,1)isgivenby
1mjκj:=1−2m−2mp.
Inthesequelwealsowrite
Fµ:=F1,µ×...×Fm,µ,g=(g1,...,gm)∈Fµ,
putfurtherewand0Eu,µ:=0W1p,µ(J;Lp(Ω;E))∩Lp,µ(J;Wp2m(Ω;E)),
0Fj,µ:=0Wκp,µj(J;Lp(Γ;E))∩Lp,µ(J;Wp2mκj(Γ;E)),0Fµ:=0F1,µ×...×0Fm,µ.
WealsowriteE0,µ(J)andE0,µ(J×Ω),andsimilarfortheotherspacesabove,ifthe
dependenceontheunderlyingintervalanddomainmightnotbeclearfromthecontext.
Asaconsequenceoftheaboveregularityassumption,(2.1.1)mightapriorinotbesolveable
foralldata(f,g,u0).Infact,forκj>1−µ+1/p,whichisequivalentto2m(µ−1/p)>
mj+1/p,itholds
Fj,µ→BUC(J;B2p,pm(µ−1/p)−mj−1/p(Γ;E)),
duetoTheorem1.3.6.Inthiscase,iftheboundaryequationin(2.1.1)holdsfort>0,by
continuityitnecessarilyalsoholdsfort=0,andthisyields
Bj(0,x,D)u0(x)=gj(0,x),x∈Γ,ifκj>1−µ+1/p.(2.1.2)
HereBju0iswell-definedforu0∈Xu,µand2m(µ−1/p)>mj+1/pprovidedthecoefficients
ofBjaresufficientlysmooth.
Thusifκj>1−µ+1/pforsomej,thenwiththeaboveapproach(2.1.1)itisnotsolvable
inEu,µforarbitrarydatag∈Fµandu0∈Xu,µ.Inthiscasethecompatibilitycondition
(2.1.2)ongandu0isnecessary.Forshort,theboundaryequationhastoholduptot=0
iftheinvolvedexpressionsarewelldefined.

68

MaximalLp,µ-RegularityforStaticBoundaryConditions

Example2.1.2.WereconsidertheproblemsfromExample2.1.1.Forthelinearized
reaction-diffusionsystem,theweightedmaximalregularityclassandtheregularityclasses
ofthedataaregivenby
Eu,µ=W1p,µ(J;Lp(Ω;E))∩Lp,µ(J;Wp2(Ω;E)),
Xu,µ=Bpp2(µ−1/p)(Ω;E),F1,µ=W1p,µ/2−1/2p(J;Lp(Γ;E))∩Lp,µ(J;Wp1−1/p(Γ;E)),
i.e.,κ1=1/2−1/2p.Compatibilityconditionsarenecessaryif2(µ−1/p)>1+1/p.
ForthelinearizedCahn-Hilliardmodelwehave
Eu,µ=W1p,µ(J;Lp(Ω;E))∩Lp,µ(J;Wp4(Ω;E)),Xu,µ=Bpp4(µ−1/p)(Ω;E),
aswellasκ1=1/4−1/4p,sothat
F1,µ=W1p,µ/4−1/4p(J;Lp(Γ;E))∩Lp,µ(J;Wp1−1/p(Γ;E)),
andfurtherκ2=3/4−1/4p,sothat
F2,µ=W3p,µ/4−1/4p(J;Lp(Γ;E))∩Lp,µ(J;Wp3−1/p(Γ;E)).
Herecompatibilityconditionsinthefirstandthesecondboundaryequationarenecessary
if4(µ−1/p)>3+1/pand4(µ−1/p)>1+1/p,respectively.
Weintendtosolve(2.1.1)inthefollowingsense.
Definition2.1.3.Wesaythattheproblem(2.1.1)enjoysthepropertyofmaximalLp,µ-
regularityontheintervalJ,iftheregularityassumptionsonthedata,i.e.,
f∈E0,µ,g∈Fµ,u0∈Xu,µ,
togetherwiththecompatibilityconditions(2.1.2),arenotonlynecessaryforaunique
solutionu∈Eu,µof(2.1.1),butalsosufficient.
TheAssumptionsontheOperators
LetP(D)=|γ|≤kpγDγbeadifferentialoperatoroforderk∈N0,withcoefficientspγ.
BythesubscriptwedenotetheprincipalpartofP,i.e.,
P(D)=pγDγ.
|γ|=k
ThesymbolofPisgivenbythepolynomialexpressionP(ξ)=|γ|≤kpγξγ,whereξ∈Rn.
Wedescribetheassumptionsonthecoefficientsoftheoperators.Itisrequiredthateach
summandoccurringinAandBjisacontinuousoperatorontherespectiveunderlying
i.e.,spaces,aαDα∈B(Eu,µ,E0,µ),|α|≤2m,(2.1.3)
furtherandbjβtrΩDβ∈B(Eu,µ,Fj,µ),|β|≤mj,j=1,...,m.(2.1.4)
Moreover,thetopordercoefficientsarerequiredtobecontinuousonJ×Ω.TheProposi-
tions1.3.16and1.3.24showthatthefollowingassumptionsaresufficientforthesepurposes.

2.1TheProblemandtheApproachinWeightedSpaces

69

(SD)For|α|<2moneofthefollowingtwoconditionsisvalid:either
2m(µ−1/p)>2m−1+n/pandaα∈E0,µ(J×Ω;B(E)),
ortherearerα,sα∈[p,∞)withp(1−sαµ)+1+2mrnα<1−2|αm|suchthat
aα∈LsαJ;(Lrα+L∞)(Ω;B(E)).
For|α|=2mitholdsaα∈BUC(J×Ω;B(E)),andifΩisunboundedtheninaddition
thelimitsaα(t,∞):=lim|x|→∞aα(t,x)existuniformlyint∈J.
(SB)Forj=1,...,mand|β|≤mjoneofthefollowingtwoconditionsisvalid:either
κj>1−µ+1/p+n−1andbjβ∈Fj,µ(J×Γ;B(E)),
mp2ortherearerjβ,sjβ∈[p,∞)with
p(1−µ)+1+n−1<κj+mj−|β|,κj+mj−|β|−(1−µ+1/p)∈/0,n−1,
sjβ2mrjβ2m2m2mrjβ
thathsucbjβ∈Bsκjjβ,pJ;Lrjβ(Γ;B(E))∩LsjβJ;Br2jβmκ,pj(Γ;B(E)).
Assuming(SB),Proposition1.3.24showthatforthetopordercoefficientsofBitholds
bjβ∈BUC(J×Γ;B(E)),|β|=mj,j=1,...,m.
Observethatthefirstconditionsin(SD),wherethecoefficientsbelongtotheweighted
spaceFj,µ,ismadeforlargep,andwillbeneededintheapplicationstoquasilinearlinear
problems.Thesecondcondition,wherebbelongstoanunweightedspace,ismadefor
lowervaluesofp,andcanbeusefulinthecontextofaprioriestimatesfortheunderlying
problem.

Weimposetwostructuralassumptionsontheoperators.Thefirstisnormalellipticity.
(E)Forallt∈J,x∈Ωand|ξ|=1itholdsσ(A(t,x,ξ))⊂C+:={Reλ>0}.IfΩis
unboundedthenitholdsinadditionσ(A(t,∞,ξ))⊂C+forallt∈Jand|ξ|=1.
ThesecondisaconditionofLopatinskii-Shapirotype.Foreachx∈Γwefixanorthogonal
matrixOν(x)thatrotatestheouterunitnormalν(x)ofΓatxto(0,...,0,−1)∈Rn,and
definetherotatedoperators(Aν,Bν)by
Aν(t,x,D):=A(t,x,OνT(x)D),Bν(t,x,D):=B(t,x,OνT(x)D).
Weassumethefollowing.

70

MaximalLp,µ-RegularityforStaticBoundaryConditions

(LS)Foreachfixedt∈Jandx∈Γ,forallλ∈C+andξ∈Rn−1with|λ|+|ξ|=0and
allh∈Emtheordinaryinitialvalueproblem
λv(y)+Aν(t,ξ,Dy)v(y)=0,y>0,
Bjν(t,ξ,Dy)v|y=0=hj,j=1,...,m,
hasauniquesolutionv∈C0([0,∞);E).1
IfEisfinitedimensional,thenitisnecessaryandsufficientfor(LS)thattheaboveinitial
valueproblemhasforh=0onlythetrivialsolution.

TheMainTheoremandtheAdvantagesoftheApproach
Themainresultofthischapterreadsasfollows.
Theorem2.1.4.LetEbeaBanachspaceofclassHT,p∈(1,∞)andµ∈(1,p,1].Let
J=(0,T)beafiniteinterval,andletΩ⊂Rnbeadomainwithcompactsmoothboundary
Γ=∂Ω.Assumethat(E),(LS),(SD)and(SB)holdtrue,andthatκj=1−µ+1/pfor
j=1,...,m.Thentheproblem
∂tu+A(t,x,D)u=f(t,x),x∈Ω,t∈J,
Bj(t,x,D)u=gj(t,x),x∈Γ,t∈J,j=1,...,m,
u(0,x)=u0(x),x∈Ω,
enjoysmaximalLp,µ-regularity,i.e.,ithasauniquesolutionu=L(f,g,u0)∈Eu,µifand
ifonly(f,g,u0)∈D:=(f,g,u0)∈E0,µ×Fµ×Xu,µ:forj=1,...,mitholds
Bj(0,∙,D)u0=gj(0,∙)onΓifκj>1−µ−1/p.
ThecorrespondingsolutionoperatorL:D→Eu,µiscontinuous.IfLisrestrictedto
D0:=(f,g,u0)∈D:g∈0Fµ,
forgivenT0>0itsoperatornormisuniformforallT≤T0.Finally,ifthecoefficients
(−i)|α|aα,|α|≤2m,(−i)|β|bjβ,|β|≤mj,j=1,...,m,(2.1.5)
andthedataarereal-valued,thenalsothesolutionuisreal-valued.
DuetoLemma1.3.25,thespacesofcompatibledataDandD0arewell-definedandBanach
spaceswhenequippedwiththenormsofE0,µ×Fµ×Xu,µandE0,µ×0Fµ×Xu,µ,respectively.
ItisimportanttodistinguishbetweenthenormsofFµand0Fµ.Theseareequivalent
forκj=1−µ+1/p,butthenormequivalentconstantsdependonthelengthofthe
underlyingintervalJ.OurmotivationtointroducethespaceD0istoobtainestimates
uniformintimeforproblemswithvanishinginitialvalues,astheytypicallyoccurinthe
1ThespaceC0([0,∞);E)consistsofthecontinuousE-valuedfunctionson[0,∞)vanishingat∞.

2.1TheProblemandtheApproachinWeightedSpaces

71

contextoffixedpointarguments(seethediscussioninRemark1.1.15).Observethatfor
(f,g,u0)∈D0itnecessarilyholdsBj(0,∙,D)u0=0onΓifthisexpressionmakessense,
i.e.,ifκj>1−µ+1/p.
Comparedtotheunweightedcase,themaximalregularityapproachinweightedspaceshas
thefollowingadvantages.
•Flexibleinitialregularity:WeobtainsolutionsforinitialvaluesinBsp,p(Ω;E),where
s∈(0,2m(1−1/p)].
•Inherentsmoothingeffect:Awayfromtheinitialtime,τ∈(0,T),thesolutionsbelong
totheunweightedspace
Eu,1(τ,T)=Wp1(τ,T;Lp(Ω;E))∩Lp(τ,T;Wp2m(Ω;E))→C(J;B2p,pm(1−1/p)(Ω;E)).
•Controlsolutionsinastrongnormatalatertimebyaweakernormatanearlier
timeandthedata:Fors=2m(µ−1/p)∈(0,2m(1−1/p)]itholds
|u(T)|B2ppm(1−1/p)(Ω;E)≤C(T)(|f|E0,µ+|g|Fµ+|u0|Bsp,p(Ω;E)).
•Avoidcompatibilityconditions:Givenp∈(1,∞),ifµissufficientlycloseto1/pthen
κj<1−µ+1/pforallj,suchthatthereisauniquesolutionu∈Eu,µforarbitrary
datainE0,µ×Fµ×Xu,µ.
ofProtheofOutlineTheproofofTheorem2.1.4isinspiredbytheoneofDenk,Hieber&Prüss[24,25]in
theunweightedcase.ThestrategyforaboundeddomainΩisasfollows,forunbounded
domainsithastobeslightlymodified.

OnedescribestheboundaryΓofΩbyafinitecollectionofcharts(Ui,ϕi),i=1,...,NH,and
furthertakesopensetsUi,i=NH+1,...,NF,suchthatUi∩Γ=∅andΩ⊂iN=1FUi.This
yieldslocalproblems,withboundaryconditionsfori=1,...,NHandwithoutboundary
conditionsfori=NH+1,...,NF.Theproblemswithoutboundaryconditionsareextended
toafull-spaceproblemandtheproblemswithboundaryconditionsaretransformedand
extendedtoahalf-spaceproblem,usingthepush-forwardcorrespondingtochartsφi.This
isdoneinSection2.4.IfthediameteroftheUiaresufficientlysmallthenbycontinuity
thetopordercoefficientsoftheresultingoperatorsareofsmalloscillation,suchthat,bya
perturbationargumentwhichisbasedonthecontractionprinciple,onecanneglectlower
ordertermsandassumethatthecoefficientsareconstant,seeSection2.3.Theresulting
full-andhalf-spaceproblemsaresolvedinSection2.2.AttheendofSection2.4,these
forΩsubordinatetothecoveri=1FUi.
solutionsareputtogethertoasoNlutionoftheoriginalproblem,usingapartitionofunity

72

MaximalLp,µ-RegularityforStaticBoundaryConditions

2.2TopOrderConstantCoefficientOperatorsonRnandR+n
2.2.1TheFull-SpaceCasewithoutBoundaryConditions
Forconstantcoefficientsaα∈B(E)weconsiderthedifferentialoperator
A(D)=aαDα.
m=2|α|ObservethattherearenolowerordertermssothatA(D)ishomogeneousofdegree2m.
Weshowthat(E)impliesparameter-ellipticity,inthesenseof[24,Definition5.1],with
angleofellipticitystrictlysmallerthanπ/2.
Lemma2.2.1.AssumethatAsatisfies(E).Thenthereisφ∈(0,π/2)suchthat
σ(A(ξ))⊂Σφ={λ∈C\{0}:|argλ|<φ},|ξ|=1,ξ∈Rn.
Proof.For|ξ|=1itholds|ξα|≤1forall|α|=2m,andtherefore
|A(ξ)|B(E)≤|aα|B(E),
m=2|α|whichyieldsthatthespectralradiusofA(ξ)isuniformlyboundedin|ξ|=1.Thusthere
isR>0,independentof|ξ|=1,suchthatλ∈ρ(A(ξ))forallλwith|λ|>Ror,
byassumption,λ∈C−.Sincetheresolventsetisopen,itfollowsfromcontinuityand
compactnessthatforallλ=iθwithθ∈[−R,R]thereisaneighbourhoodUiθ⊂Cofiθ
suchthatUiθ⊂ρ(A(ξ))forall|ξ|=1.Againcompactnessyieldsaradiusr>0,which
doesnotdependonθ∈[−R,R],suchthatBr(iθ)⊂ρ(A(ξ))forall|ξ|=1.Wethusobtain
anangleφ∈(0,π/2)withρ(A(ξ))⊃C\Σφ.
WehavethefollowingmaximalLp,µ-regularityresultforAonthehalf-line.
Proposition2.2.2.LetEbeaBanachspaceofclassHT,p∈(1,∞),µ∈(1/p,1],and
assumethatAsatisfies(E).Thenthereisauniquesolutionu=SF(f,u0)∈Eu,µ(R+×Rn)
of

u+∂tu+A(D)u=f(t,x),x∈Rn,t>0,
u(0,x)=u0(x),x∈Rn,(2.2.1)
ifonlyandiff∈E0,µ(R+×Rn),u0∈Xu,µ(Rn).
ThecorrespondingsolutionoperatorSF:E0,µ(R+×Rn)×Xu,µ(Rn)→Eu,µiscontinuous.
Proof.ItfollowsfromLemma2.2.1thatAisparameter-elliptic,withangleofellipticity
strictlysmallerthanπ/2.Thusby[24,Theorem5.5]andtheperturbationresult[24,Propo-
sition2.11],therealizationof1+AonLp(Rn;E)withdomainD(1+A)=Wp2m(Rn;E)
isinvertibleandadmitsaboundedH∞-calculuswithH∞-anglestrictlysmallerthanπ/2.

2.2TopOrderConstantCoefficientOperatorsonRnandR+n

73

Itisnowaconsequenceof(A.3.2)and[85,Theorem4.2]that1+Aenjoysmaximal
Lp-regularityonthehalf-line,i.e.,1+A∈MRpR+;Lp(Rn;E).Since
Xu,µ(Rn)=B2p,pm(µ−1/p)(Rn;E)=Lp(Rn;E),Wp2m(Rn;E)µ−1/p,p
byPropositionA.4.2,theassertionfollowsfromTheorem1.2.3byPrüss&Simonett.

2.2.2TheHalf-SpaceCasewithBoundaryConditions
Forconstantcoefficientsaα,bjβ∈B(E)wenowconsidertheoperators
A(D)=aαDα,Bj(D)=bjβtrR+nDβ,j=1,...,m.
|α|=2m|β|=mj
Observethatagaintherearenolowerorderterms.WeidentifytheboundaryofR+nwith
Rn−1.NowallspacesmustbeunderstoodoverR+×R+nandR+×Rn−1,respectively,i.e.,
Eu,µ=W1p,µ(R+;Lp(R+n;E))∩Lp,µ(R+;Wp2m(R+n;E)),
Fj,µ=Wκp,µj(R+;Lp(R+n;E))∩Lp,µ(R+;Wp2mκj(R+n;E)),
E0,µ=Lp,µ(R+;Lp(R+n;E)),Xu,µ=B2p,pm(µ−1/p)(R+n;E).
TheBanachspaceofcompatibledataisgivenby
D=(f,g,u0)∈E0,µ×Fµ×Xu,µ:forj=1,...,mitholds
Bj(D)u0=gj(0,∙)onΓifκj>1−µ+1/p.
Themainresultofthissubsectionisthefollowing.
Proposition2.2.3.LetEbeaBanachspaceofclassHT,p∈(1,∞),µ∈(1/p,1],and
assumethat(A,B)satisfies(E)and(LS).Supposefurtherthatκj=1−µ+1/pforall
j=1,...,m.Thereisauniquesolutionu=SH(f,g,u0)∈Eu,µfortheproblem
u+∂tu+A(D)u=f(t,x),x∈R+n,t>0,
Bj(D)u=gj(t,x),x∈Rn−1,t>0,j=1,...,m,(2.2.2)
u(0,x)=u0(x),x∈R+n,
ifandonlyif(f,g,u0)∈D.ThesolutionoperatorSH:D→Eu,µiscontinuous.
AsexplainedinSection2.1,thenecessaryconditionsonthedataareaconsequenceof
themappingbehaviourofspatialderivatives,thespatialtraceandthetemporaltraceon
theweightedanisotropicspaces,derivedinLemma1.3.4,Proposition1.3.12andTheorem
1.3.6.Ifasolutionoperatorexists,thenitscontinuityfollowsfrom
1+∂t+A(D)∈B(Eu,µ,E0,µ),B(D)∈B(Eu,µ,Fµ),trt=0∈B(Eu,µ,Xu,µ)
andtheopenmappingtheorem.

74

MaximalLp,µ-RegularityforStaticBoundaryConditions

Ourtaskisthustoshowthatforanygiven(f,g,u0)∈Dtheproblem(2.2.2)hasaunique
solutionu∈Eu,µ.Forthiswefollowthestrategypresentedin[25,Section4].Wefirst
consider(2.2.2)inLemma2.2.5withhomogeneousboundaryconditions,g=0,andthen
weconsider(2.2.2)inLemma2.2.6withf=0andu0=0.Thegeneralcasefollowsfrom
acombinationoftheselemmasandwillbeshownattheendofthissubsection.
Asfornormalellipticity,wefirstshowthatalso(LS)holdsinfactonalargersectorthan
assumed.originallyLemma2.2.4.Let(A,B)satisfy(E)and(LS).Thenthereisφ∈(0,π/2)suchthatfor
allλ∈Σπ−φandξ∈Rn−1with|λ|+|ξ|=0andallh=(h1,...,hm)∈Emtheordinary
problemaluevinitialλv(y)+A(ξ,Dy)v(y)=0,y>0,
Bj(ξ,Dy)v|y=0=hj,j=1,...,m,(2.2.3)
hasauniquesolutionv∈C0([0,∞);E),i.e.,thecondition(LS)for(A,B)isevenvalidfor
λ∈Σπ−φ.
Proof.(I)ItfollowsfromLemma2.2.1thatAhasangleofellipticityφA∈(0,π/2).For
λ∈Σπ−φAandξ∈Rn−1werewritetheordinarydifferentialequationλv+A(ξ,Dy)v=0
oforder2mtoasystemof2mfirstorderequations,
∂yv(y)=iA0(λ,ξ)v(y),y>0,v=(v,∂yv,...,∂y2m−1v),
whereA0(λ,ξ)isaB(E)-valued2m×2m-matrix.Thesolutionsoftheaboveequationare
oftheformv(y)=eyiA0(λ,ξ)v0,wherev0∈E2m.
By[24,Proposition6.1],thematrixiA0(λ,ξ)hasaspectralgapattheimaginaryaxis.
WedenotetheprojectionontothestablepartofthespectrumbyPs(λ,ξ)∈B(E2m).
Denotingfurtherbyπ1:E2m→Ethecanonicalprojectionontothefirstcomponent,we
definetheoperatorpencilT:Σπ−φA×Rn−1→BPs(λ,ξ)E2m,Emby
T(λ,ξ)v0:=B(ξ,Dy)π1eyiA0(λ,ξ)v0y=0,v0∈Ps(λ,ξ)E2m.
Forλandξfromacompactset,thespectralgapforiA(λ,ξ)isuniform,andPsis
continuousinitsarguments.Byconstruction,(2.2.3)isuniquelysolveableforλ∈Σπ−φA
andξ∈Rn−1ifandonlyifT(λ,ξ)isinvertible.
(II)Letvbetheuniquesolutionof(2.2.3)inC0([0,∞);E)forh∈Em.Thenforr>0
satisfiesalsovfunctiontheλv(ry)+A(ξ,Dy)v(ry)=0,y>0,
Bj(ξ,Dy)v|y=0=hj,j=1,...,m.
Since(Dyv)(r∙)=r−1Dy(v(r∙)),itfollowsfromhomogeneitythatw:=v(r∙)istheunique
ofsolutionr2mλw(y)+A(rξ,Dy)w(y)=0,y>0,
Bj(rξ,Dy)w|y=0=rmjhj,j=1,...,m.

2.2TopOrderConstantCoefficientOperatorsonRnandR+n

75

ThereforeT(λ,ξ)isinvertibleifandonlyifT(r2mλ,rξ)isinvertible.
(III)By(LS),continuityandcompactnessthereisanangleφ∈(φA,π/2)suchthat
T(λ,ξ)isinvertibleforall
(λ,ξ)∈{se±iθ:s∈[0,1],θ∈[π/2,π−φ]}×{|ξ|=1},
allforfurtherand(λ,ξ)∈{e±iθ:θ∈[π/2,π−φ]}×{|ξ|≤1}.
WeusethisfactandthescalingpropertyfromStepIItoshowthatT(λ,ξ)isinvertible
forallλ∈Σπ−φandξ∈Rn−1with|λ|+|ξ|=0.Wedistinguishfourcases.
For1≤|λ|≤|ξ|=:r1theoperatorT(λ/r12m,ξ/r1)isinvertiblebecause|λ|/r12m≤1
and|ξ|/r1=1.ThescalingpropertythusshowsthatT(λ,ξ)isinvertibleinthiscase,
andhenceitisinvertiblewhenever|λ|=1.For1≤|ξ|≤|λ|=:r22mtheoperator
T(λ/r22m,ξ/r2)isinvertibledueto|λ|/r22m=1.SoT(λ,ξ)isinvertibleif|λ|,|ξ|≥1.
Nowfor0<r32m:=|λ|≤1andarbitraryξwehavethatT(λ/r32m,ξ/r3)isinvert-
iblebecause|λ|/r32m=1.Finally,for0<r4:=|ξ|≤1andarbitraryλtheoperator
T(λ/r42m,ξ/r4)isinvertiblebecause|ξ|/r4=1.
Forhomogeneousboundaryconditions,weightedmaximalregularityfollowsagainfrom
theunweightedcase,sincetheabstractresultof[71]isapplicable.
Lemma2.2.5.LetEbeaBanachspaceofclassHT,p∈(1,∞),µ∈(1/p,1],andassume
that(A,B)satisfies(E)and(LS).Thenforallf∈E0,µandu0∈Xu,µthereisaunique
solutionu∈Eu,µof
u+∂tu+A(D)u=f(t,x),x∈R+n,t>0,
Bj(D)u=0,x∈Rn−1,t>0,j=1,...,m,(2.2.4)
u(0,x)=u0(x),x∈R+n.
DenotingbyABtherealizationoftheoperatorAonLp(R+n;E),withdomain
D(AB)={u∈Wp2m(R+n;E):Bu=0},
theoperator1+ABgeneratesanexponentiallystableanalyticC0-semigroup,and1+AB∈
MRp,µ(R+;Lp(R+n;E)).
Proof.DuetotheLemmas2.2.1and2.2.4,theoperatorAisparameterellipticwith
angleofellipticityφA<π/2,andforφ∈(φA,π)itholdsthat(A,B)satisfies(LS)forall
λ∈Σπ−φ.Thus,by[24,Theorem7.4]andtheperturbationresult[24,Proposition2.11],
1+ABisinvertible,andadmitsaboundedH∞-calculuswithH∞-anglestrictlysmaller
thanπ/2.Itfollowsfrom(A.3.2),[85,Theorem4.2]andTheorem1.2.3that1+AB∈
MRp,µ(R+;Lp(R+n;E)).Since
Xu,µ=Lp(R+n;E),Wp2m(Rn;E)µ−1/p,p


76

MaximalLp,µ-RegularityforStaticBoundaryConditions

byPropositionA.4.2weobtaintheuniquesolvabilityof(2.2.4)inEu,µ,forf∈E0,µand
u0∈Xu,µ.Inparticular,1+ABisthegeneratorofananalyticsemigroup.
Itseemsnotpossibletoabsorbinhomogeneousboundaryconditions,g=0,intothedomain
ofareasonableoperatoronLp,µ(R+;E).Henceinthiscasewecannotreducemaximal
Lp,µ-regularitytotheunweightedproblemviatheabstractresultofTheorem1.2.3.
Totreattheinhomogeneousboundaryconditions,wefirstconsideranellipticproblem
correspondingto(2.2.2).ThefollowingresultisacombinationoftheLemmas4.3and4.4
[25].inLemma2.2.6.LetEbeaBanachspaceofclassHT,p∈(1,∞)andassumethat(A,B)
satisfies(E)and(LS).Thenforλ∈C+\{0}andgj∈Wp2mκj(Rn−1;E),j=1,...,m,the
problemλv+A(D)v=0,x∈R+n,
Bj(D)v=gj(x),x∈Rn−1,j=1,...,m,(2.2.5)
hasauniquesolutionv(λ)∈Wp2m(R+n;E).Thissolutionmayberepresentedintheform
mv(λ)=Sj(λ)gj,
=1jforoperatorsSj(λ)∈B(Wp2mκj(Rn−1;E),Wp2m(R+n;E))givenby
Sj(λ)=Tj(λ)Lλ1−mj/2mEλ.
HereLλ:=λ+(−Δn−1)m,andtheextensionoperatorEλ=e−∙Lλ1/2mmapsgj∈
Wp2mκj(Rn−1;E)tothefunction(x,y)→e−yLλ1/2mgj(x),withx∈Rn−1andy>0.
Moreover,forσ≥0and|α|≤2mitholdsDαTj(σ+i∙)∈C1R\{0};B(Lp(R+n;E)),and
λ1−2|αm|DαTj(λ),λ2−2|αm|∂DαTj(λ):λ=σ+iθ∈C+\{0},|α|≤2m,j=1,...,m
θ∂isanR-boundedsetofoperatorsinB(Lp(R+n;E)).
Withtheaboveresultthetime-dependentproblemwithinhomogeneousboundarycon-
ditionscannowbesolvedviaFouriertransformwithrespecttotime,usingtheabove
representationofthesolutionsofthecorrespondingstationaryproblems.Recallthenota-
tion0E1,µ=0W1p,µ(R+;Lp(R+n;E))∩Lp,µ(R+;Wp2m(R+n;E)),
0Fj,µ=0Wκp,µj(R+;Lp(Rn−1;E))∩Lp,µ(R+;Wp2mκj(Rn−1;E)),
0Fµ=0F1,µ×...×0Fm,µ.
Lemma2.2.7.LetEbeaBanachspaceofclassHT,p∈(1,∞),µ∈(1/p,1],andassume
that(A,B)satisfies(E),(LS).Thenforg∈0Fµthereisauniquesolutionu∈0Eu,µof
u+∂tu+A(D)u=0,x∈R+n,t>0,
Bj(D)u=gj(t,x),x∈Rn−1,t>0,j=1,...,m,(2.2.6)
u(0,x)=0,x∈R+n.

2.2TopOrderConstantCoefficientOperatorsonRnandR+n

77

Proof.Throughoutwewritex=(x,y)∈R+nwithx∈Rn−1andy>0.
(I)ItfollowsfromLemma2.2.5thatsolutionsu∈0Eu,µof(2.2.6)areunique.Forthe
existenceofasolutionwearegoingtoconstructasolutionoperator
L:Cc∞(R+;Wp2m(Rn−1;E))m→0Eu,µ,
andshowthatitadmitstheestimate
|Lg|Eu,µ|g|0Fµ,g∈Cc∞(R+;Wp2m(Rn−1;E))m.(2.2.7)
ByLemma1.3.14,thesetCc∞(R+;Wp2m(Rn−1;E))misdensein0Fµ(notethat2mκj∈/N0).
Hence,if(2.2.7)holds,thenLextendstoacontinuousoperator0Fµ→0Eu,µ.Since
1+∂t+A(D)∈B(0Eu,µ,E0,µ),B(D)∈B(0Eu,µ,0Fµ),
thefunctionu=Lgisthentheuniquesolutionof(2.2.6)forg∈0Fµ.
(II)ToconstructthesolutionoperatorL,letg∈Cc∞(R+;Wp2m(Rn−1;E))m.Inthesequel
weidentifysuchafunctionwithitstrivialtemporalextensiontoR.ApplyingtheFourier
transformFtwithrespecttot∈Rto(2.2.6),anddenotingthecovariablebyθ∈R,we
arriveforeachθatthestationaryproblem
(1+iθ)v+A(D)v=0,x∈R+n,
Bj(D)v=(Ftgj)(θ,x),x∈Rn−1,j=1,...,m.(2.2.8)
ByLemma2.2.6,theuniquesolutionv(θ)∈Wp2m(R+n;E)of(2.2.8)isgivenby
mv(θ)=Tj(1+iθ)L11+−iθmj/2mE1+i∙Ft(gj)(θ),
=1jwhereL1+iθ=1+iθ+(−Δn−1)m,andwhereforθ∈RtheextensionoperatorE1+i∙isfor
h∈Lp,µ(R+;Lp(Rn−1;E))definedby
(E1+iθh)(t,x,y):=e−yL11+/i2θmh(t,x),t∈R,(x,y)∈R+n.
Dueto[24,Corollary1.9],forθ∈Randy>0wehavetherepresentation
L1−mj/2me−yL11+/i2θm=1z1−mj/2me−yz1/2m(z−L1+iθ)−1dz,
1+iθ2πiΞ
whereΞ=(∞,δ]ei3π/2∪δei[3π/2,−3π/2]∪[δ,∞)e−i3π/2forsomesufficientlysmallδ>0.
Thusforeachy>0theB(Lp(Rn−1;E))-valuedfunction
θ→L11+−iθmj/2me−yL11+/i2θm,θ∈R,
issmoothandallofitsderivativesarebounded.Sinceθ→Ft(gj)(θ)israpidlydecreasing
andTj(1+i∙)isbyLemma2.2.6auniformlyboundedfamilyofoperators,itholdsthatthe
solutionvof(2.2.8)israpidlydecreasinginθ.WemaythereforeapplytheinverseFourier
transformtov,andobtainthat
mu=Lg:=Ft−1Tj(1+i∙)L11+−i∙mj/2mE1+i∙Ftgj
=1j

78

MaximalLp,µ-RegularityforStaticBoundaryConditions

solvesthedifferentialequationsin(2.2.6).Toshowu(0)=0,wefirstobservethatu(0)∈
D(AB)holdssinceuissmoothintwithvaluesinWp2m(R+n;E)andsatisfiestheequations.
functiontheHenceu=u−e−∙(1+AB)u(0)
satisfiesu(0)=u(0)=0,whichyields
uR∈C1R;Lp(R+n;E)∩CR;Wp2m(R+n;E)
forthetrivialextensionuRofutoR.Further,asthesemigroupgeneratedby1+ABis
exponentiallystable,thefunctionsuanduarerapidlydecreasingonR+.Thus(FtuR)(θ)
solves(2.2.8)foreachθ∈R.ByuniquenessitholdsFtu=Ftu,andthereforeu=u,
whichyieldsu(0)=0.
(III)Toshowtheestimate(2.2.7)wederiveanotherrepresentationforL.W1/e2mhaveseen
abovethatforgj∈Cc∞(R;Wp2m(Rn−1;E))thefunctionθ→L11+−iθmj/2me−yL1+iθFtgjbe-
longstotheSchwartzclass.HenceFourierinversionholds,andwemaywrite
mLg:=Ft−1Tj(1+i∙)FtFt−1L11+−i∙mj/2mE1+i∙Ftgj.
=1jOnS(R;Wp2m(Rn−1;E))itholdsFt−1L1+iθ=LFt−1,with
L:=1+∂t+(−Δn−1)m.
Moreover,byLemma1.3.1therealizationofLonLp,µ(R+;Lp(Rn−1;E))withdomain
D(L)=0W1p,µ(R+;Lp(Rn−1;E))∩Lp,µ(R+;Wp2m(Rn−1;E))
isinvertibleandsectorialofanglenotlargerthanπ/2.Using[24,Corollary1.9]forL1+iθ
andL,weobtainfory>0andgj∈Cc∞(R;Wp2m(Rn−1;E))
Ft−1L11+−i∙mj/2me−yL11+/i2∙mFtgj=1z1−mj/2me−yz1/2m(z−L)−1Ft−1Ftgjdz
iπ2Ξ=L1−mj/2me−yL1/2mgj.
DenotingbyEtheextensionoperator
Eh(t,x,y):=e−yL1/2mh(t,x),t>0,(x,y)∈Rn+,h∈Lp,µ(R+;Lp(Rn−1;E)),
wearriveattherepresentation
mLg=Ft−1Tj(1+i∙)FtL1−mj/2mEgjforg∈Cc∞(R+;Wp2m(Rn−1;E))m.
=1j(IV)DuetoLemma1.3.8,theoperatorEmapscontinuously
0Fj,µ=DL1/2m(2m−mj−1/p,p)→Lp(R+;DL1/2m(2m−mj,p)),

79

2.3TopOrderCoefficientshavingSmallOscillation79
andL1−mj/2m=L1/2m2m−mjiscontinuous
Lp(R+;DL1/2m(2m−mj,p))→Lp(R+;Lp,µ(R+;Lp(Rn−1;E)))=E0,µ.
Further,duetoLemma2.2.6andTheorem1.2.4,foreachj=1,...,mand|α|≤2mthe
B(Lp(R+n;E))-valuedsymbolDαTj(1+i∙)isaFouriermultiplieronLp,µ.Thereforethe
operatorFt−1Tj(1+i∙)Ftiscontinuous2
E0,µ→Lp,µ(R+;Wp2m(R+n;E)).
Finally,itfollowsfromtheequation∂tu=−(1+A(D))uthattheEu,µ-normofucanbe
controlledbyitsLp,µ(R+;Wp2m(R+n;E))-norm,whichshows(2.2.7).
Theexistenceofauniquesolutionof(2.2.2)forgiven(f,g,u0)∈Disnowaconsequence
oftheLemmas2.2.5and2.2.7,asfollows.Denotebyu1∈Eu,µthesolutionof
w+∂tw+A(D)w=f(t,x),x∈R+n,t>0,
Bj(D)w=0,x∈Rn−1,t>0,j=1,...,m,
w(0,x)=u0(x),x∈R+n,
whichexistsbyLemma2.2.5.SinceBj(D)u0=gj|t=0forκj>1−µ+1/p,itfollowsfrom
Proposition1.1.11andκj=1−µ+1/pthat3
gj−Bj(D)u1∈0Fj,µ,j=1,...,m.
Ifwedenotebyu2∈Eu,µthesolutionof
w+∂tw+A(D)w=0,x∈R+n,t>0,
Bj(D)w=gj(t,x)−Bj(D)u1(t,x),x∈Rn−1,t>0,j=1,...,m,
w(0,x)=0,x∈R+n,
whichexistsbyLemma2.2.7,thenu=u1+u2solves(2.2.2).Theuniquenessofthis
solutionfollowsfromtheuniquenessofsolutionsof(2.2.4).Finally,thecontinuityofthe
solutionoperatorSHof(2.2.2)isaconsequenceofthefactthatDisaBanachspaceand
theopenmappingtheorem.ThusProposition2.2.3isestablished.

2.3TopOrderCoefficientshavingSmallOscillation
Fromnowonwerestrictourconsiderationstoafinitetimeinterval
J=(0,T),T>0.
2Proceedingasintheproofof[25,Lemma4.4],onecanshowthatforj=1,...,mand|α|≤2mit
holdsDαTj(1+i∙)∈C2(R;B(Lp(R+n;E))),andthat|∂θ2DαTj(1+iθ)|θ12.HencealsoProposition1.2.5
applies.3Atthispointwehavetoexcludethevalueκj=1−µ+1/p.

80

MaximalLp,µ-RegularityforStaticBoundaryConditions

Wefirstconsiderthehalf-spacecase,andwrite
Eu,µ(J)=Eu,µ(J×R+n),Fµ(J)=Fµ(J×Rn−1),
andsoon.LettheoperatorsAandBj,j=1,...,m,begivenby
A(t,x,D)=aα(t,x)Dα,t∈J,x∈R+n,
m2|≤α|andBj(t,x,D)=bjβ(t,x)trR+nDβ,t∈J,x∈Rn−1.
m|≤β|jObservethat,incontrasttotheprevioussection,theoperatorsmayhavelowerorderterms,
andtheB(E)-valuedcoefficientsaαandbjβareallowedtodependon(t,x).
Thetopordercoefficientsoftheboundaryoperatorsareassumedtobeoftheform
aα(t,x)=aα0+aα(t,x),|α|=2m,(2.3.1)
bjβ(t,x)=bj0β+bjβ(t,x),|β|=mj,j=1,...,m,(2.3.2)
whereaα0,bj0β∈B(E)donotdependon(t,x).Usingthemwedefineauxiliarytoporder
constantcoefficientoperators(A0,B0)by
A0(D):=aα0Dα,Bj0(D):=bj0βtrR+nDβ,j=1,...,m.(2.3.3)
|α|=2m|β|=mj
Assuming(SD)and(SB)forthecoefficientsofA−A0andB−B0,thePropositions1.3.16
thatensure1.3.24and

A∈B(Eu,µ(J),E0,µ(J)),B∈B(Eu,µ(J),Fµ(J)).(2.3.4)
Moreover,(SD)and(SB)imply
aα∈BUC(J×R+n;B(E)),|α|=2m,
bjβ∈BUC(J×Rn−1;B(E)),|β|=mj,j=1,...,m.
ForanintervalJ=(0,T)withT>0thesetofcompatibledataisgivenby
D(J)=(f,g,u0)∈E0,µ(J)×Fµ(J)×Xu,µ:forj=1,...,mitholds
Bj(0,∙,D)u0=gj(0,∙)onRn−1ifκj>1−µ+1/p,
consideralsoewandD0(J)=(f,g,u0)∈D(J):g∈0Fµ(J).
DuetoLemma1.3.25,theseareBanachspacesasclosedsubspacesofE0,µ(J)×Fµ(J)×
Xu,µandE0,µ(J)×0Fµ(J)×Xu,µ,respectively.Wehavethefollowingresultforthe
half-space.

2.3TopOrderCoefficientshavingSmallOscillation

81

Proposition2.3.1.LetEbeaBanachspaceofclassHT,p∈(1,∞),andµ∈(1/p,1].
Assumethat(A0,B0)satisfies(E)and(LS),andthatthecoefficientsof(A−A0,B−B0)
satisfy(SD)and(SB).Supposefurtherthatκj=1−µ+1/pforj=1,...,m.Thenthere
areatimeT0∈(0,T]andanumberε>0suchthatif
(t,x)∈[0sup,T0]×Rn|aα(t,x)|B(E)<ε,|α|=2m,(2.3.5)
+and(t,x)∈[0,Tsup0]×Rn−1|bjβ(t,x)|B(E)<ε,|β|=mj,j=1,...,m,(2.3.6)
thenforeachintervalJ=(0,T)withT∈(0,T0]thereisauniquesolutionu=
SHsm(f,g,u0)∈Eu,µ(J)of
∂tu+A(t,x,D)u=f(t,x),x∈R+n,t∈J,
Bj(t,x,D)u=gj(t,x),x∈Rn−1,t∈J,j=1,...,m,(2.3.7)
u(0,x)=u0(x),x∈R+n,
ifandonlyif(f,g,u0)∈D(J).Thesolutionoperator
SHsm:D(J)→Eu,µ(J)
iscontinuous.RestrictedtoD0(J),itsoperatornormisindependentofT∈(0,T0].
Proof.Throughoutthisproof,let0<T≤T0≤T,andsetJ0=(0,T0).
(I)Wefirstconsiderthenecessitypart.Letu∈Eu,µ(J)beasolutionof(2.3.7).Then
(2.3.4)yieldsf∈E0,µ(J)andg∈Fµ(J),andTheorem1.3.6impliesu0∈Xu,µ.Hence
(f,g,u0)∈D(J)isnecessarytoobtainasolutionu∈Eu,µ(J).
(II)NowsupposethatforeachT∈(0,T0]itholdsthatforall(f,g,u0)∈D(J)thereis
auniquesolutionu∈Eu,µ(J)of(2.3.7),i.e.,thereisasolutionoperatorSHsmfor(2.3.7).
ThenSHsmiscontinuousdueto(2.3.4)andtheopenmappingtheorem.Fromthisabstract
argumentitsoperatornormdependsonT∈(0,T0](ourconstructionbelowdoesnot
removethisdependence,see(2.3.10)).
For(f,g,u0)∈D0(J)wemayextendf∈E0,µ(J)andg∈0Fµ(J)toEJ0f∈E0,µ(R+)
andEJ0g∈0Fµ(R+),respectively,usingtheextensionoperatorEJ0fromLemma1.1.5,
whosenormisindependentofT.Ofcourse,thenitholds(EJ0f|J0,EJ0g|J0,u0)∈D(J0),
anditfollowsfromtheassumeduniquenessofsolutionsof(2.3.7)that
SHsm(f,g,u0)=SHsm(EJ0f|J0,EJ0g|J0,u0)|J.
obtainoretherefeW|SHsm(f,g,u0)|Eu,µ(J)|EJ0f|E0,µ(R+)+|EJ0g|0Fµ(R+)+|u0|Xu,µ
|f|E0,µ(J)+|g|0Fµ(J)+|u0|Xu,µ,
wheretheconstantsinthisestimateonlydependonT0,butnotonT∈(0,T0].

82

MaximalLp,µ-RegularityforStaticBoundaryConditions

(III)Itremainstofindauniquesolutionu∈Eu,µ(J)of(2.3.7)forgiven(f,g,u0)∈D(J).
defineeWZu0(J):=v∈Eu,µ(J):v(0,∙)=u0,
whichisanonemptyclosedsubspaceofEu,µ(J)duetoLemma1.3.9.Forgivenv∈Zu0(J)
weconsidertheproblem
w+∂tw+A0w=f+(A0−A+1)vinJ×R+n,
B0w=g+(B0−B)vonJ×Rn−1,(2.3.8)
w(0,∙)=u0inR+n,
wherethetoporderconstantcoefficientoperatorsA0andB0aregivenby(2.3.3).Dueto
Lemma1.2.1,solutionsof(2.3.8)areuniqueinEu,µ(J)forv∈Zu0(J),sincebyLemma
2.2.5therealizationof1+AB00onLp(Rn+;E)isthegeneratorofananalyticC0-semigroup.
Tofindasolutionw=S(v)∈Eu,µ(J)of(2.3.8)weconsidertheproblem
w+∂tw+A0w=fonR+×R+n,
B0w=gonR+×Rn−1,(2.3.9)
w(0,∙)=w0onR+n.
Since(A0,B0)areassumedtosatisfy(E)and(LS),Proposition2.2.3yieldsacontinuous
eratoropsolutionSH:DB0(R+)→Eu,µ(R+)
for(2.3.9),whereDB0(R+)denotesthespaceofcompatibledatawithrespecttoB0.Since
gandu0arecompatiblewithrespecttoB,itfollowsthat
(EJ(f+(A0−A+1)v),EJ(g+(B0−B)v),u0)∈DB0(R+),
whereEJistheextensionoperatorfromJtoR+,seeLemma1.1.5.Therefore
w=S(v):=SHEJ(f+(A0−A+1)v),EJ(g+(B0−B)v),u0|J(2.3.10)
istheuniquesolutionof(2.3.8).Observethatafunctionu∈Eu,µ(J)solves(2.3.7)ifand
onlyifitisafixedpointofSinZu0(J).
(IV)WeshowthatShasauniquefixedpointinZu0(J)viathecontractionprinciple,
providedT0andthusthelengthofJaresufficientlysmall.ClearlySmapsZu0(J)into
itself.Forv1,v2∈Zu0(J),thedifferenceS(v1)−S(v2)solves
w+∂tw+A0w=(A0−A+1)(v1−v2)onJ×R+n,
B0w=(B0−B)(v1−v2)onJ×Rn−1,(2.3.11)
w(0,∙)=0onR+n.
From(v1−v2)(0,∙)=0weinferthat(B0−B)(v1−v2)∈0Fµ(J).Wethusmayextend
thedataT-independentlytoR+,usingEJ0fromLemma1.1.5.Sincetherestrictionofthe

2.3TopOrderCoefficientshavingSmallOscillation

83

problemhalf-linetheofsolutionw+∂tw+A0w=EJ0(A0−A+1)(v1−v2)onR+×Rn+,
B0w=EJ0(B0−B)(v1−v2)onR+×Rn−1,
w(0,∙)=0onR+n,
toJsolves(2.3.11),itfollowsfromtheuniquenessofsolutionsof(2.3.11)that
S(v1)−S(v2)=SHEJ0(A0−A+1)(v1−v2),EJ0(B0−B)(v1−v2),0|J.
ThecontinuityofSHandEJ0nowyield
0000|S(v1)−S(v2)|Eu,µ(J)≤|SH(EJ(A−A+1)(v1−v2),EJ(B−B)(v1−v2),0)|Eu,µ(R+)
0000|EJ(A−A+1)(v1−v2)|E0,µ(R+)+|EJ(B−B)(v1−v2)|0Fu,µ(R+)
00|(A−A+1)(v1−v2)|E0,µ(J)+|(B−B)(v1−v2)|0Fµ(J),(2.3.12)
wheretheconstantinthisestimateisindependentofT0.
oldshIt(V)|(A0−A+1)(v1−v2)|E0,µ(J)≤|aαDα(v1−v2)|E0,µ(J)
m=2|α|α+|aαD(v1−v2)|E0,µ(J)+|v1−v2|E0,µ(J).
|α|<2m
Forthefirstsummandassumption(2.3.5)yields
α|aαD(v1−v2)|E0,µ(J)ε|v1−v2|Eu,µ(J).
m=2|α|Forthesecondsummandand|α|<2m,supposethatthesecondconditionin(SD)holds.
Thenwetakeδ∈p(1−µ)+1+n,1−|α|andapplyLemma1.3.15onpointwisemul-
sα2mrα2m
obtaintotipliers,α|aαD(v1−v2)|E0,µ(J)
|α|<2m
α|aα|Lsα(J;Lrα(R+n;B(E)))|D(v1−v2)|0Hδp,µ(J;Lp(R+n;E))∩Lp,µ(J;Hp2mδ(R+n;E)).
|α|<2m
Itfollowsfrom(v1−v2)(0,∙)=0andLemma1.3.13thatforgivenη>0wehave
α|D(v1−v2)|0Hδp,µ(J;Lp(R+n;E))∩Lp,µ(J;Hp2mδ(R+n;E))≤η|v1−v2|Eu,µ(J),
providedT0issufficientlysmall.Ifthelowerordercoefficientssatisfythefirstcondition
αin(SD)oneobtainsthisestimatefor|α|<2m|aαD(v1−v2)|E0,µ(J)inasimilarway.For
thethirdsummandwehave
|v1−v2|E0,µ(J)≤η|v1−v2|Eu,µ(J).

84

MaximalLp,µ-RegularityforStaticBoundaryConditions

Combiningtheseinequalities,wearriveat
|(A0−A+1)(v1−v2)|E0,µ(J)(ε+η)|v1−v2|Eu,µ(J).
(VI)Wenowestimatetheboundarytermsin(2.3.12).Forj=1,...,mitholds
|(Bj0−Bj)(v1−v2)|0Fj,µ(J)≤|bjβtrR+nDβ(v1−v2)|0Fj,µ(J)(2.3.13)
|β|=mj
+|bjβtrR+nDβ(v1−v2)|0Fj,µ(J).
<m|β|jFor|β|=mjweuse(SB),theLemmas1.3.21,1.3.22,1.3.23and(2.3.6),toestimatewith
δ∈(0,κj)
|bjβtrR+nDβ(v1−v2)|0Fj,µ(J)ε|trR+nDβ(v1−v2)|0Fj,µ(J)(2.3.14)
+|bjβ|Y(J)|trR+nDβ(v1−v2)|0Wδp,µ,2mδ(J×R+n;E)).
HereY(J)=Fj,µ(J×Rn−1;B(E))orY(J)=Bsκjjβ,p(Lrjβ)∩Lsjβ(Br2jβmκ,pj),accordingtothe
twoconditionsin(SB).Ifthefirstconditionin(SB)isvalidonefurtherhastousethe
eddingbem0Wδp,µ,2mδ(J×R+n;E))→BUC(J×R+n;E)
todeduce(2.3.14)fromLemma1.3.23,whichisvalidforsomeδ∈(0,κj)ifκj>1−µ+
1/p+2n−mp1.Notealsothat(2.3.14)isuniforminT≤T0dueto(v1−v2)(0,∙)=0.Forthe
firstsummand,weinferfromProposition1.3.12,Lemma1.3.4andthat|bjβ|Y(J)arefixed
ersbumn|trR+nDβ(v1−v2)|0Fj,µ(J)|v1−v2|Eu,µ(J),
wherethisestimateisagainuniformin|J|≤T0.ForthesecondsummandweuseLemma
1.3.13toobtainforgivenη
|bjβ|Y(J)|trR+nDβ(v1−v2)|0Wδp,µ,2mδ(J×R+n;E))≤η|v1−v2|Eu,µ(J),
providedT0issufficientlysmall.Thisyields
|bjβtrR+nDβ(v1−v2)|0Fj,µ(J)(ε+η)|v1−v2|Eu,µ(J)
|β|=mj
forthefirstsummandin(2.3.13).Forthesecondsummandin(2.3.13)and|β|<mjwe
useinasimilarway(SB)andtheLemmas1.3.13,1.3.21,1.3.23and1.3.22,toobtain
|bjβtrR+nDβ(v1−v2)|0Fj,µ(J)≤η|v1−v2|Eu,µ(J)
<m|β|jforsufficientlysmallT0.Itisthusshownthat
|(Bj0−Bj)(v1−v2)|0Fj,µ(J)(ε+η)|v1−v2|Eu,µ(J).

2.3TopOrderCoefficientshavingSmallOscillation

85

(VII)Comparingwith(2.3.12)andchoosingεandη,i.e.,T0,sufficientlysmall,weobtain
thatSisastrictcontraction,andthereforehasauniquefixedpointinZu0(J).
Wenowturntothefullspaceproblem,andwrite
Eu,µ(J)=Eu,µ(J×Rn),E0,µ(J)=E0,µ(J×Rn),Xu,µ=Xu,µ(Rn).
WeconsidertheoperatorAonRnwithB(E)-valuedvariablecoefficientsaα,givenby
A(t,x,D)=aα(t,x)Dα,(t,x)∈J×Rn.
m2|≤α|Wehavethefollowingresult.
Proposition2.3.2.LetEbeaBanachspaceofclassHT,p∈(1,∞),andµ∈(1/p,1].
AssumethatAsatisfies(E)and(SD).ThenthereareatimeT0∈(0,T]andnumberε>0
ifthathsucsupn|aα(t,x)−aα(0,0)|B(E)<ε,|α|=2m,(2.3.15)
(t,x)∈[0,T0]×R
thenforeachintervalJ=(0,T)withT∈(0,T0]thereisauniquesolutionu=
SFsm(f,u0)∈Eu,µ(J)of
∂tu+A(t,x,D)u=f(t,x),x∈Rn,t∈J,(2.3.16)
u(0,x)=u0(x),x∈Rn,
ifandonlyif(f,u0)∈E0,µ(J)×Xu,µ.Thesolutionoperator
SFsm:E0,µ(J)×Xu,µ→Eu,µ(J)
iscontinuous,anditsoperatornormisindependentofT∈(0,T0].
Proof.Theproofiscompletelyanalogoustothehalf-spacecase.Welet0<T≤T0≤T.
AsintheproofofProposition2.3.1weobtainthenecessaryconditionsonthedata.To
showthatfor(f,u0)∈E0,µ(J)×Xu,µauniquesolutionof(2.3.16)exists,weconsiderthe
spaceZu0(J)=v∈Eu,µ(J):v(0,∙)=u0,
andforv∈Zu0(J)theproblem
w+∂tw+A0w=f+(A0−A+1)vonJ×Rn,(2.3.17)
w(0,∙)=u0onRn.
HeretheoperatorA0isgivenbyA0:=|α|=2maα(0,0)Dα.Theuniquesolutionof(2.3.17)
ybengivisw=S(v):=SFEJ0(f+(A0−A+1)v),u0|J,
whereSF:E0,µ(R+)×Xu,µ→Eu,µ(R+)isthecontinuoussolutionoperatorfor(2.3.17)
onR+×RnfromProposition2.2.2.AsintheproofofProposition2.3.1onecanshow
thatSisastrictcontractiononZu0(J),providedεandT0aresufficientlysmall.The

86

MaximalLp,µ-RegularityforStaticBoundaryConditions

resultinguniquefixedpointu∈Eu,µ(J)ofSistheuniquesolutionof(2.3.16).Using
thatu=SFEJ0(f+(A0−A+1)u),u0|JandemployingthecontinuityofSFweobtain
thatthenormofthesolutionoperatorSFsmisindependentofT,sincethenormofEJ0is
it.oftendenindep

2.4TheGeneralCaseonaDomain
InthissectionwefinallyproveTheorem2.1.4.LetEbeaBanachspaceofclassHT,let
J=(0,T)beafiniteinterval,andletΩ⊂Rnbeadomainwithcompactsmoothboundary
Γ=∂Ω.Nowwewrite
Eu,µ=Eu,µ(J×Ω),Fµ=Fµ(J×Γ),
andsoon.Weconsidertheproblem
∂tu+A(t,x,D)u=f(t,x),x∈Ω,t∈J,
Bj(t,x,D)u=gj(t,x),x∈Γ,t∈J,j=1,...,m,(2.4.1)
u(0,x)=u0(x),x∈Ω,
wherethedifferentialoperatorsAandBj,j=1,...,m,aregivenby
A(t,x,D)=aα(t,x)Dα,t∈J,x∈Ω,
m2|≤α|Bj(t,x,D)=bjβ(t,x)trΩDβ,t∈J,x∈Γ,mj∈{0,...,2m−1}.
m|≤β|jTheB(E)-valuedcoefficientsaαandbjβareassumedtosatisfy(SD)and(SB).Inthiscase
thePropositions1.3.16and1.3.24ensurethat
A∈B(Eu,µ,E0,µ),B∈B(Eu,µ,Fµ).(2.4.2)
Moreover,itisincludedinresp.followsfromtheseassumptionsthatthetopordercoeffi-
satisfytscienaα∈BUC(J×Ω;B(E)),|α|=2m,
bjβ∈BUC(J×Γ;B(E)),|β|=mj,j=1,...,m.
Thesetofcompatibledataisgivenby
D=(f,g,u0)∈E0,µ×Fµ×Xu,µ:forj=1,...,mitholds
Bj(0,∙,D)u0=gj(0,∙)onΓifκj>1−µ+1/p,
furtherandD0=(f,g,u0)∈D:g∈0Fµ.
Thefollowinglocalizationprocedureisverylong,elaborate,andlookssophisticated,but
afterallitisnothingbutasequenceofsimpleprinciples,andalotofnotation.Foran
outlinewerefertotheendofSection2.1.

2.4TheGeneralCaseonaDomain

87

2.1.4.TheoremofofPro(I)ThenecessaryconditionsonthedatafollowasinSection2.1from(2.4.2)andTheorem
1.3.6.IfthesolutionoperatorLexists,thenitscontinuityanditsdependenceonthelength
ofJcanbeshownasinStepIIoftheproofofProposition2.3.1.
Ifthecoefficientsoftheoperatorsareasin(2.1.5)andifthedataisreal-valued,thenwe
havethatifu∈Eu,µsolves(2.4.1),thenalsoReu∈Eu,µsolves(2.4.1).Henceu=Reuif
oneassumesuniqueness,i.e.,thesolutionisreal-valuedinthiscase.
(II)Given(f,g,u0)∈D,wehavetoshowthatthereexistsauniquesolutionu∈Eu,µof
(2.4.1).WefirstshowthatitsufficestoobtainthisundertheassumptionthatT=|J|is
l.smaltlysufficienUsingtheextensionoperatorEJfromLemma1.1.5wemayassumethatthecoefficientsof
AandBaredefinedon[0,2T].SupposethatforeachT∗∈[0,2T)wecanfinda(small)
timeτT∗∈(0,2T−T∗)suchthattheproblem
∂tu+A(t,x,D)u=f(t,x),x∈Ω,t∈(T∗,T∗+τT∗),
B(t,x,D)u=g(t,x),x∈Γ,t∈(T∗,T∗+τT∗),(2.4.3)
u(T∗,x)=u0(x),x∈Ω,
hasauniquesolutionu∈Eu,µ(T∗,T∗+τT∗)forall
f∈E0,µ(T∗,T∗+τT∗),g∈Fµ(T∗,T∗+τT∗),u0∈Xu,µ,
whichsatisfythecompatibilitycondition
Bj(T∗,∙,D)u0=gj(0,∙),onΓ,ifκj>1−µ+1/p,j=1,...,m.
Inthiscasewecansolve(2.4.1)uniquelyforgiven(f,g,u0)∈D(J)asfollows.UsingEJ
wemayassumethatalsofandgaredefinedon(0,2T).Thesolutionintervalsfor(2.4.3)
yieldanopencoverof[τ0,T],fromwhichwechooseafinitesubcoverkK=1(Tk,Tk+τk)
withT1<τ0,Tk<Tk−1+τk−1for1<k≤K,T<Tk+τk.
Letu0∈Eu,µ(0,τ0)betheuniquesolutionof(2.4.3)on(0,τ0)withdata
f=f|(0,τ0),g=g|(0,τ0),u0=u0.
SinceT1<τ0,u0(T1)∈Xu,µ4andthecompatibilityconditionholds,thereisaunique
solutionu1∈Eu,µ(T1,T1+τ1)of(2.4.3)on(T1,T1+τ1)withdata
f=f|(T1,T1+τ1),g=g|(T1,T1+τ1),u0=u0(T1,∙).
Sinceweassumethatsolutionsof(2.4.3)areuniqueforallinitialtimesT∗∈[0,2T),
itfollowsthatu0andu1coincideon(T1,τ0).Iteratingthisargumentyieldsfunctions
uk∈Eu,µ(Tk,Tk+τk),k=1,...,K,suchthatuksatisfies(2.4.3)on(Tk,Tk+τk)withdata
f=f|(Tk,Tk+τk),g=g|(Tk,Tk+τk),u0=uk−1(Tk,∙),
4Infact,duetotheinherentsmoothingeffectoftheweightedspacesitevenholdsu0(T1)∈Xu,1.

88

MaximalLp,µ-RegularityforStaticBoundaryConditions

andsuchthatukanduk+1coincideon(Tk+1,Tk+τk).Sincetheweightonlyhasaneffect
attheinitialtimesTk,itholds
uk|(Tk+1,Tk+τk)=uk+1|(Tk+1,Tk+τk)∈Eu,1(Tk+1,Tk+τk).
Hencewemayputtogetherthefunctionsuk,k=0,...,K,toafunctionu∈Eu,µ(0,TK+
τK),thatsolves(2.4.1)onJ=(0,T).Ourassumptionalsoimpliesthatthissolutionis
unique.Observethattherestrictionof(A,B)toanysubintervalofJ=(0,T)isstillsubjectto
(E),(LS),(SD)and(SB).Therefore,duetotheaboveconsiderations,ourobjectiveisto
showtheuniquesolvabilityof(2.4.1)forall(f,g,u0)∈D(J),undertheassumptionthat
T=|J|issufficientlysmall.
(III)WeintendtousethePropositions2.3.1and2.3.2toshowuniquesolvability.Tothis
endwehavetolocalize(2.4.1)alsoinspace.IfΩisunboundedwechoosealargenumber
R>0withΓ⊂BR(0)andset
x0:=∞,U0:=Ω\BR(0).
WedefineonJ×Rnextendedtopordercoefficientsaα0=aα0(t,x),|α|=2m,by
aα(t,x),x∈U0,
aα0(t,x):=aαt,R2|xx|2,x∈BR(0)\{0},(2.4.4)
aα(t,x0),x=0,
andfurtheronJ×Rnextendedlowerordercoefficientsaα0=aα0(t,x),|α|<2m,by
a0(t,x):=aα(t,x),x∈U0,
α0,x∈BR(0).
Usingthesecoefficientswedefinethedifferentialoperator
A0(t,x,D):=aα0(t,x)Dα.
m2|≤α|Observethatfor|α|=2mthefunctionsaα0arecontinuousextensionsoftheaαtoRn,
whichonlyusevaluesofaα|U0.ThereforeA0satisfies(E),sincethisisapointwisecondition.
Moreover,byassumption(SD),thelimitaα(t,∞)=lim|x|→∞aα(t,x)existsuniformlyin
t∈J,|α|=2m.Thus,givenε>0,ifRissufficientlylargeandTissufficientlysmallthen
bycontinuityitholds
t∈J,supx∈U0|aα(t,x)−aα(0,x0)|B(E)<ε,|α|=2m.
Byconstruction,thiscarriesovertotheextendedtopordercoefficients,
00t∈J,xsup∈Rn|aα(t,x)−aα(0,0)|B(E)<ε,|α|=2m.
Hence,duetoProposition2.3.2,ifRislargeandTissmallthenforallJ=(0,T)with
T≤Tthereisacontinuoussolutionoperator
SFsm,0:E0,µ(J×Rn)×Xu,µ(Rn)→Eu,µ(J×Rn),

2.4TheGeneralCaseonaDomain

89

(2.4.6)(2.4.7)

problemfull-spacethefor∂tv+A0(t,x,D)v=f0(t,x),x∈Rnt∈J,(2.4.5)
v(0,x)=u00(x),x∈Rn.
(IV)Now,ifΩisunbounded,takeapointx∗∈Ω\U0=Ω∩BR(0),andifΩisbounded,
takeapointx∗∈Ω.InbothcasesweconstructadifferentialoperatorAx∗onRnasabove.
Choosearadiusrx∗>0with
Brx∗(x∗)∩Γ=∅,
putandUx∗:=Brx∗(x∗).
Wedefineextendedtopordercoefficientsaαx∗=aαx∗(t,x),|α|=2m,by
xaα(t,x),x∈Ux∗,
aα∗(t,x):=aαt,x∗+rx2∗|xx−−xx∗|2,x∈/Ux∗,(2.4.6)
∗extendedlowerordercoefficientsaαx∗=aαx∗(t,x),|α|<2m,by
aαx∗(t,x):=aα(t,x),x∈Ux∗,(2.4.7)
0,x∈/Ux∗,
andwefinallyset
Ax∗(t,x,D):=aαx∗(t,x)Dα.
m2|≤α|Asabove,theoperatorAx∗satisfies(E),andifrx∗andTaresufficientlysmall,then
Proposition2.3.2yieldsthatforallT∈≤Tthereisacontinuoussolutionoperator
SFsm,x∗:E0,µ(J×Rn)×Xu,µ(Rn)→Eu,µ(J×Rn),
whereJ=(0,T),forthefull-spaceproblem
∂tv+Ax∗(t,x,D)v=f∗(t,x),x∈Rnt∈J,(2.4.8)
v(0,x)=u0∗(x),x∈Rn.
(V)Forapointx∗∈Γ=∂ΩwechooseanopenneighbourhoodUx∗ofx∗inRnsuch
thattherearesmoothdiffeomorphismsϕx∗:Ux∗→Rnandaradiusrx∗>0withthe
ertiespropϕx∗(x∗)=0,ϕx∗(Ux∗)=B2rx∗(0),ϕx∗(x∗)=Oν(x∗),
ϕx∗(Ux∗∩Ω)⊂R+n,ϕx∗(Ux∗∩Γ)⊂Rn−1.(2.4.9)
NotethatweidentifyRn−1withRn−1×{0}⊂Rn.FurtherOν(x∗)istheorthogonalmatrix
fixedinassumption(LS)thatrotatestheouterunitnormalν(x∗)to(0,...,0,−1)∈Rn.By
LemmaA.1.1,achart(Ux∗,ϕx∗)withtheabovepropertiesalwaysexists.Wemayassume

90

MaximalLp,µ-RegularityforStaticBoundaryConditions

thatthesup-normsofanyderivativeofϕx∗andϕx−∗1areuniformlyboundedforrx∗≤1.
setfurthereWUx∗:=ϕx−∗1Brx∗(0),
anddenotebyΦx∗thepush-forwardoperatorcorrespondingtoϕx∗,i.e.,Φx∗v=v◦ϕx−∗1.
NowwedefinethedifferentialoperatorAΦx∗forfunctionsv:R+n∩Brx∗(0)→Eby
AΦx∗(t,x,D)v:=Φx∗A(t,∙,D)Φx−∗1v(x),t∈J,x∈R+n∩Brx∗(0),
andtheboundaryoperatorsBjΦx∗,j=1,...,m,by
BjΦx∗(t,x,D)v:=Φx∗Bj(t,∙,D)Φx−∗1v(x),t∈J,x∈Rn−1∩Brx∗(0),j=1,...,m.
Forv∈Eu,µR+n∩Brx∗(0)and1≤|α|≤2mitholds
Dα(Φx−∗1v)(t,x)=qαγ(x)(Dγv)(t,ϕx∗(x)),t∈J,x∈Ω∩Ux∗,
1≤|γ|≤|α|
whereqαγarereal-valuedboundedsmoothfunctionsinx,dependingonthepartialderiva-
tivesofthecomponentsofϕx∗(see[69,Section1.1.7]).Thusthepushedoperatorsare
formtheofagainAΦx∗(t,x,D)=aαΦx∗(t,x)Dα,
|α|≤2m
BjΦx∗(t,x,D)=bjΦβx∗(t,x)trR+nDβ,j=1,...,m,
m|≤β|jwheretrR+ndenotesthespatialtraceoperatorforR+n.LemmaA.1.2impliesthattheprin-
cipalpartsof(AΦi,BΦi)aregivenby
AΦi(t,x,D)=A(t,x,OνT(x∗)D),BΦi(t,x,D)=B(t,x,OνT(x∗)D).(2.4.10)
DuetoLemmaA.4.1,andsincethefunctionsqαγaresmoothandbounded,thecoefficients
aαΦx∗satisfy(SD),formulatedforJ×R+n∩Brx∗(0),andthecoefficientsbjΦβx∗satisfy(SB)
onJ×Rn−1∩Brx∗(0).
(VI)WenowextendthetopordercoefficientsofAΦx∗fromR+n∩Brx∗(0)toR+nby
reflectionasin(2.4.6),andthelowerordercoefficientsofAΦx∗triviallyfromR+n∩Brx∗(0)
toR+nasin(2.4.7).Denotingtheextendedcoefficientsbyaαx∗,thisyieldsanoperator
Ax∗(t,x,D):=aαx∗(t,x)Dα,t∈J,x∈R+n.
m2|≤α|WefurtherdefinethetoporderconstantcoefficientoperatorAx∗,0by
Ax∗,0(D):=Ax∗(0,0,D)=aαx∗,0Dα,aαx∗,0:=aαx∗(0,0)=aα(0,x∗).
m=2|α|Itfollowsfrom(2.4.10)thatforξ∈Rnitholds
Ax∗,0(ξ)=A(0,x∗,OνT(x∗)ξ).

2.4TheGeneralCaseonaDomain

91

SinceAsatisfies(E)wethusobtainthatAx∗,0satisfies(E)aswell.Wewritethetoporder
coefficientsofAx∗intheform
aαx∗(t,x)=aαx∗,0+aαx∗(t,x),|α|=2m,
whereaαx∗(t,x)=aαx∗(t,x)−aαx∗,0,asrequiredfor(2.3.1).Byconstruction,thecoefficientsof
Ax∗−Ax∗,0satisfy(SD).Givenε>0,ifT,rx∗andthediameterofUx∗aresufficientlysmall,
thenthecoefficientsaαΦx∗haveoscillationlessthanεaroundaαΦx∗(0,0)onJ×R+n∩Brx∗(0)
forall|α|=2m.Byconstructionwehave
nR×Jsup|aαx∗|B(E)<ε,|α|=2m,
+forthetopordercoefficientsofAx∗aswell.ToextendthetopordercoefficientsofBjΦi
fromRn−1∩Brx∗(0)toRn−1tocoefficientsbjxβ∗wefixanonnegativecut-offfunction
χ∈Cc∞(Rn−1)with
χ(x)=1,|x|≤1,χ(x)=0,|x|≥2,χ(x)∈[0,1],x∈Rn−1,
andsetfor|β|=mjandj=1,...,m
bjxβ∗(t,x):=bjΦβx∗(0,0)+χ(x/2rx∗)bjΦβx∗(t,χ(x/rx∗)x)−bjΦβx∗(0,0),t∈J,x∈Rn−1.
(2.4.11)ThelowerordercoefficientsofBjΦiareextendedonRn−1tocoefficientsbjxβ∗bysetting
bjxβ∗:=ERn−1∩Brx∗(0)bjΦβx∗,|β|<mj,j=1,...,m,(2.4.12)
whereERn−1∩Brx∗(0)denotesthespatialextensionoperatorfromRn−1∩Brx∗(0)toRn−1,
givenby(1.3.3).Theseextendedcoefficientsyieldboundaryoperators
Bjx∗(t,x,D):=bjxβ∗(t,x)trR+nDβ,t∈J,x∈Rn−1,j=1,...,m.
m|≤β|jWedefinethetoporderconstantcoefficientoperatorBx∗,0=(B1x∗,0,...,Bxm∗,0)by
Bjx∗,0(D):=bjxβ∗,0trR+nDβ,bjxβ∗,0:=bjxβ∗(0,0)=bjβ(0,x∗),j=1,...,m.
|β|=mj
Dueto(2.4.10),forξ∈Rn−1wehavethat
Ax∗,0(ξ,Dy)=A0,x∗,OνT(x∗)(ξ,Dy),Bx∗,0(ξ,Dy)=B0,x∗,OνT(x∗)(ξ,Dy).
Nowweseethattheassumption(LS)for(A,B)onΩisjustmadethat(Ax∗,0,Bx∗,0)
satisfies(LS)onR+n.WewritethetopordercoefficientsofBx∗intheform
bjxβ∗=bjxβ∗,0+bjxβ∗,|β|=mj,j=1,...,m,
asrequiredfor(2.3.2).ByconstructionwehavethatthecoefficientsofBx∗−Bx∗,0satisfy
(SB).AsforthetopordercoefficientsofAx∗,forgivenε,ifT,rx∗andthediameterofUx∗
aresufficientlysmallthenitfollowsfromcontinuitythat
J×Rn−1
sup|bjxβ∗|B(E)<ε,|β|=mj,j=1,...,m.

92

MaximalLp,µ-RegularityforStaticBoundaryConditions

Therefore(Ax∗,Bx∗)satisfiesalltheassumptionsofProposition2.3.1,andifεandTare
small,thenforallJ=(0,T)withT≤Tthereisacontinuoussolutionoperator
SHsm,x∗:DBx∗(J)→Eu,µ(J)
problemhalf-spacethefor∂tv+Ax∗(t,x,D)v=f∗(t,x),x∈R+n,t∈J,
Bjx∗v=gj∗(t,x),x∈Rn−1,t∈J,j=1,...,m,(2.4.13)
v(0,x)=u0∗(x),x∈R+n.
Here,DBx∗(J)denotesthesetofcompatibledata(f∗,g∗,u∗0)withrespecttoBx∗.
(VII)ThesetsUx∗,togetherwithU0ifΩisunbounded,yieldanopencoverofΩ.If
Ωisboundeditfollowsfromcompactnessthattherearefinitelymanypointsxi∈Ω,
i=1,...,NFforsomeNF∈N,andfinitelymanypointsxi∈Γ,i=NF+1,...,NHfor
someNH>NF,suchthattheunionofthecorrespondingsets
Ui:=Uxi,i=1,...,NH,
coversΩ.IfΩisunbounded,weobtaininthesamewayafinitecoverforthecompactset
Ω\U0.SettingU0:=∅ifΩisbounded,wethusobtaininanycaseafinitecover
NNHFΩ⊂Ui∪Ui,(2.4.14)
i=0i=NF+1
togetherwithcorrespondingpointsxi,operatorsAifori=0,...,NFand(Ai,Bi)for
i=NF+1,...,NH.IfεandTaresmall,thentherearesolutionoperators
SFsm,i,i=0,...,NF,andSHsm,i,i=NF+1,...,NH,
forthefinitelymanyfull-andhalf-spaceproblems(2.4.8)and(2.4.13)onJ=(0,T),
correspondingtoAiand(Ai,Bi),respectively.
(VIII)IfΩisboundedthereexistsapartitionofunity{ψi}i=1,...,NHforΩ,subordinate
tothecover(2.4.14).Intheunboundedcasethereissuchapartitionforthecompactset
Ω\U0.Thuswesetinadditionψ0:=0intheboundedcase,and
NHψ0:=1−ψi
=1iintheunboundedcase,suchthat{ψi}i=0,...,NHisinanycaseapartitionofunityforΩ,
(2.4.14).tordinateosubNowtakecompatibledata(f,g,u0)∈D(J)for(A,B),andconsidertheproblem(2.4.1),
forwhichwehavetoshowuniquesolvability.Supposethatu∈Eu,µ(J×Ω)solves(2.4.1).
Thenusolvesthelocalizedproblems
∂t(ψiu)+A(ψiu)=ψif+[A,ψi]uinΩ∩Ui,t∈J,
B(ψiu)=ψig+[B,ψi]uonΓ∩Ui,t∈J,(2.4.15)
(ψiu)(0,∙)=ψiu0inΩ∩Ui,

2.4TheGeneralCaseonaDomain

93

foreachi=0,...,NH.Here[∙,∙]denotesthecommutatorbracket,e.g.,
[A,ψi]u=A(ψiu)−ψiAu.
Observethat[A,ψi]and[Bj,ψi]aredifferentialoperatorsoflowerorder,i.e.,lessorequal
than2m−1andmj−1,respectively.Fori=0,...,NFitholdsΩ∩Ui=Ui,sothatthere
arenoboundaryconditionsinvolvedin(2.4.15)inthiscase.BytheconsiderationsinStep
IV,thefunctionψiuistheuniquesolutionoftheinitial-valueproblem
∂tv+Ai(t,x,D)v=fi(t,x),x∈Rn,t∈J,
v(0,x)=u0i(x),x∈Rn,
wherewehaveset
fi:=ψif+[A,ψi]u,u0i:=ψiu0,i=0,...,NF,
andwhereweidentifyfunctionswithcompactsupportwiththeirtrivialextensiontoRn.
holdsthereforeItψiu=SFsm,i(fi,u0i;u)|Ui,i=0,...,NF.
HerethenotationSFsm,i(fi,ui0;u)indicatesthatfiisdefinedwithrespecttou.
Fori=NF+1,...,NHwehaveΓ∩Ui=∅,sothatboundaryconditionsareinvolvedin
(2.4.15)inthiscase.Wetransform(2.4.15)toaflatboundary,usingthepushforwardΦi
correspondingtoϕi.Thenv=Φi(ψiu)satisfies
∂tv+Ai(t,x,D)v=fi(t,x),x∈R+n,t∈J,
Bi(t,x,D)v=gi(t,x),x∈Rn−1,t∈J,
v(0,x)=u0i(x),x∈R+n,
wherethistimewehaveset,fori=NF+1,...,NH,
fi:=Φi(ψif+[A,ψi]u),gi(t,x)=Φi(ψig+[B,ψi]u),u0i:=Φi(ψiu0),
identifyingfunctionswiththeirtrivialextensiontoR+nasabove.Bytheconsiderationsin
StepVanduniquenessitholds
ψiu=Φi−1SHsm,i(fi,gi,u0i;u)|R+n∩Bri(0),i=NF+1,...,NH,
whereagainthenotationSHsm,i(fi,gi,u0i;u)indicatesthatfi,giaredefinedwithrespectto
u.Noteherethat(fi,gi,u0i)iscompatiblewithrespecttoBi,since(f,g,u0)iscompatible
withrespecttotheoriginalboundaryoperatorB.
(IX)Wechoosescalar-valuedfunctionsφi∈Cc∞(Rn),i=0,...,NH,suchthat
φi≡1onsuppψi,suppφi⊂Ui.
TheniN=0Hφiψi≡1onΩ.For(f,g,u0)∈D(J)weconsidertheBanachspace
Zu0(J):=u∈Eu,µ(J×Ω):u(0,∙)=u0,


94

MaximalLp,µ-RegularityforStaticBoundaryConditions

whichisnonemptybyLemma1.3.9,anddefineonZu0(J)themapGby
Gf,g,u0(u):=φiSFsm,i(fi,u0i;u)|Ui+φiΦi−1(SHsm,i(fi,gi,u0i;u)|R+n∩Bri(0)),
NFNH
i=0i=NF+1
wherefi,giandu0iaredefinedasabovewithrespecttou,respectively.Bytheconsidera-
tionsinthelaststep,forasolutionu∈Eu,µ(J)of(2.4.1)itholds
NFGf,g,u0(u)=φiψiu=u.
=0iThusasolutionof(2.4.1)hastobeafixedpointofGf,g,u0inZu0(J).
UsingthecontractionprincipleweshowthatGf,g,u0hasauniquefixedpointinZu0(J)
forallcompatibledata(f,g,u0)∈D(J),providedTissufficientlysmall.Byconstruction,
Gf,g,u0isaselfmappingonZu0(J).Fori=0,...,NFandu1,u2∈Zu0(J),thefunction
v=SFsm,i(fi,1,u0i;u1)−SFsm,i(fi,2,u0i;u2)
istheuniquesolutionof
∂tv+Ai(t,x,D)v=[A,ψi](u1−u2)inRn,t∈J,
v(0,∙)=0inRn,
where[A,ψi](u1−u2)isidentifiedwithitstrivialspatialextensiontoRn.Itthereforeholds
v=SFsm,i[A,ψi](u1−u2),0.
ByProposition2.3.2,theoperatornormofSFsm,iisindependentofT.Givenη>0,weuse
thisfact,that[A,ψi]isoflowerorder,thatthecoefficientsofAaresubjectto(SD),and
that(u1−u2)(0,∙)=0,todeducefromLemma1.3.13theestimate
|φiSFsm,i(fi,1,u0i;u1)−φiSFsm,i(fi,2,u0i;u2)|Eu,µ(J×Ω)
|SFsm,i([A,ψi](u1−u2),0)|Eu,µ(J×Rn)
|[A,ψi](u1−u2)|E0,µ(J×Ω)
≤η|u1−u2|E1,µ(J×Ω),
providedTissufficientlysmall.Similarly,fori=NF+1,...,NHthefunction
v=SHsm,i(fi,1,gi,1,u0i;u1)−SHsm,i(fi,2,gi,2,u0i;u2)
istheuniquesolutionof
∂tv+Ai(t,x,D)v=Φi([A,ψi](u1−u2))inR+n,t∈J,
Bi(t,x,D)v=Φi([B,ψi](u1−u2))onRn−1,t∈J,
v(0,∙)=0inR+n,

2.4TheGeneralCaseonaDomain

95

whereweagainidentifytheright-handsideswiththeirtrivialextensionstoR+nandRn−1,
respectively.Noteherethattherequiredcompatibilityconditionatt=0holds,dueto
(u1−u2)(0,∙)=0.Therefore
v=SHsm,iΦi([A,ψi](u1−u2)),Φi([B,ψi](u1−u2)),0.
ByProposition2.3.1,theoperatornormofSHsm,irestrictedtovanishinginitialvaluesis
uniforminTsmallerthanagivenlength.Usingthesametoolsasabove,togetherwith
theLemmas1.3.21,1.3.22and1.3.23aboutpointwisemultiplicationontheboundary,we
engivforobtainη|φiΦi−1SHsm,i(fi,1,gi,1,u0i;u1)|R+n∩Bri(0)−SHsm,i(fi,2,gi,2,u0i;u2)|R+n∩Bri(0)|Eu,µ(J×Ω)
|SHsm,i(Φi([A,ψi](u1−u2)),Φi([B,ψi](u1−u2)),0)|Eu,µ(J×R+n)
|[A,ψi](u1−u2)|E0,µ(J×Ω)+|[B,ψi](u1−u2)|0Fµ(J×Γ)
≤η|u1−u2|Eu,µ(J×Ω),
providedTissufficientlysmall.HenceforsmallTthemapGf,g,u0isastrictcontractionon
Zu0(J)andhasauniquefixedpointinthere.Sincethisholdstrueforall(f,g,u0)∈D(J),
thisfactalreadyimpliesthatsolutionsof(2.4.1)areunique.Wefurtherobtainalinear
maptoinpfixed

Q:D(J)→Zu0(J),Q(f,g,u0)=Gf,g,u0(Q(f,g,u0)).
Wedefinethespace
D00(J):={(f,g,0)∈D0(J)},
andusetheaboveestimatesandthecontinuityofthesolutionoperatorsSFsm,iandSHsm,i
obtainto

|Q(f,g,0)|Eu,µ(J)≤|Gf,g,0(Q(f,g,0))−Gf,g,0(0)|Eu,µ(J)+|Gf,g,0(0)|Eu,µ(J)
η|Q(f,g,0)|Eu,µ(J)+|(f,g,0)|D0(J)
for(f,g,0)∈D00(J),whereηissmall.HencetheoperatornormofQ:D00(J)→Z0(J)is
uniforminTsmallerthanagivenlength.Notethat,duetothenonemptyintersectionsof
theUi,thefunctionQ(f,g,u0)doesnotsolve(2.4.1)withright-handside(f,g,u0)∈D(J),
general.in(X)Weconstructasolutionfor(2.4.1)byfindingforgiven(f,g,u0)∈D(J)theappropri-
ate(f,g,u0)∈D(J)forwhichQ(f,g,u0)solves(2.4.1).Inotherwords,overlappings
inthesuminthedefinitionofGf,g,u0comingfromnonemptyintersectionsoftheUihave
tocancelintherightway.AsQmapsintoZu0(J)itisclearthatwemusthaveu0=u0.
Solet(f,g,u0)∈D(J)begiven.For(f,g,u0)∈D(J)weusethatQ(f,g,u0)isthe

96

MaximalLp,µ-RegularityforStaticBoundaryConditions

fixedpointofGf,g,u0,toobtain
NF(∂t+A)Q(f,g,u0)=(∂t+A)φiSFsm,i(f,i,u0i;Q(f,g,u0))|Ui
=0iNH+(∂t+A)φiΦi−1SHsm,i(f,i,g,i,u0i;Q(f,g,u0))|Rn∩Br(0))
i=NF+1+i
NH=f+K1(f,g)+φi[A,ψi]Q(f,g,u0),
=0itermncorrectiothewithNFK1(f,g):=[A,φi]SFsm,i(f,i,u0i;Q(f,g,u0))|Ui
=0iNH+[A,φi]Φi−1SHsm,i(f,i,g,i,u0i;Q(f,g,u0))|Rn∩Br(0).
i=NF+1+i
Notethat,since{ψi}isapartitionofunityforΩandφi≡1onsuppψi,itholds
NHφi[A,ψi]Q(f,g,u0)=[A,1]Q(f,g,u0)=0.
=0iSimilarly,ontheboundarywehave
BQ(f,g,u0)=g+K2(f,g),
termcorrectionthewithK2(f,g):=[B,φi]Φi−1SH(f,i,g,i,u0i;Q(f,g,u0))|R+n∩Bri(0).
NHsm,i
+1N=iFHerethetermsinvolvingSFsm,idonotappearsincethefunctionsφivanishonΓfori=
0,...,NF.Itfollowsthatthedesired(f,g)isasolutionoftheequation
(f,g)+(K1,K2)(f,g)=(f,g).(2.4.16)
SinceQ(f,g,u0)|t=0=u0forκj>1−µ+1/pitholds
NHNH
K2(f,g)j|t=0=[Bj(0,∙,D),φi]ψiu0=[Bj(0,∙,D),1]ψiu0=0.
i=NF+1i=NF+1
ThereforeK2mapsinto0Fµ(J).Inordertosolve(2.4.16)wethusconsidertheequation
(f,g)+(K1,K2)(f,g)=−(K1,K2)(f,g).(2.4.17)
for(f,g)∈E0,µ(J)×0Fµ(J).Ifwecanfindasolution(f,g)of(2.4.17)then
(f,g)=(f+f,g+g)

2.4TheGeneralCaseonaDomain

97

solves(2.4.16)andsatisfies(f,g,u0)∈D(J),whichfinishestheproof.
(XI)Weshowthat(2.4.17)hasauniquesolution(f,g)∈E0,µ(J)×0Fµ(J),usingonce
morethecontractionprinciple.Wehavealreadyseenthat
(f,g)→−(K1,K2)(f,g)−(K1,K2)(f,g)(2.4.18)
isaselfmaponE0,µ(J)×0Fµ(J).Toshowthatitisacontraction,take(f1,g1),(f2,g2)∈
E0,µ(J)×0Fµ(J),put
u1:=Q(f1,g1,u0),u2:=Q(f2,g2,u0),
considerand|(K1,K2)(f1,g1)−(K1,K2)(f2,g2)|E0,µ(J)×0Fµ(J)(2.4.19)
=|K1(f1,g1)−K1(f2,g2)|E0,µ(J)+|K2(f1,g1)−K2(f2,g2)|0Fµ(J).
Forthefirstsummandweusethat[A,φi]isoflowerorder,toobtainwithLemma1.3.13
forgivenη>0
NF|K1(f1,g1)−K1(f2,g2)|E0,µ(J×Ω)≤η|SFsm,i(f1,i,u0i;u1)−SFsm,i(f2,i,u0i;u2)|Eu,µ(J×Rn)
=0iNH+η|SHsm,i(f1,i,g1,i,u0i;u1)−SHsm,i(f2,i,g2,i,u0i;u2)|Eu,µ(J×Rn),(2.4.20)
++1N=iFprovidedTissufficientlysmall.Weconcentrateonthesecondsumin(2.4.20).Fori=
NF+1,...,NHthefunctionv=SHsm,i(f1,i,g1,i,u0i;u1)−SHsm,i(f2,i,g2,i,u0i;u2)solves
∂tv+Aiv=Φiψi(f1−f2)+[A,ψi](u1−u2)inR+n,t∈J,
Biv=Φiψi(g1−g2)+[B,ψi](u1−u2)onRn−1,t∈J,
v(0,∙)=0inR+n.
Wemaythusestimate
|SHsm,i(f1,i,g1,i,u0i;u1)−SHsm,i(f2,i,g2,i,u0i;u2)|Eu,µ(J×R+n)
|f1−f2|E0,µ(J×Ω)+|[A,ψi](u1−u2)|E0,µ(J×Ω)
+|g1−g2|0Fµ(J×Γ)+|[B,ψi](u1−u2)|0Fµ(J×Γ)
|f1−f2|E0,µ(J×Ω)+|g1−g2|0Fµ(J×Γ)+|Q(f1−f2,g1−g2,0)|Eu,µ(J×Rn)
+|f1−f2|E0,µ(J×Ω)+|g1−g2|0Fµ(J×Γ)
uniformlyinT,usingthattheoperatornormofSHsm,ionthe0Fµ-spaces,ofQ:D00(J)→
Z0(J),ofthespatialtraceandthespatialderivativesareuniforminT,respectively.Sim-
ilarlyoneestimatesthefirstsumin(2.4.20)uniformlyinT,toobtainforgivenη
|K1(f1,g1)−K1(f2,g2)|E0,µ(J×Ω)≤η|(f1,g1)−(f2,g2)|E0,µ(J×Ω)×Fµ(J×Γ),
providedTissufficientlysmall.Inthesamewayoneshowsthecorrespondingestimatefor
thetermin(2.4.19)whereK2isinvolved.Thusthemap(2.4.18)isastrictcontraction
onE0,µ(J)×0Fµ(J),andtheresultingfixedpointistheuniquesolutionof(2.4.17).This
finallyprovesTheorem2.1.4.

98

MaximalLp,µ-RegularityforStaticBoundaryConditions

(2.5.1)(2.5.2)

2.5ARight-InversefortheBoundaryOperator
Inthissectionweconstructaright-inverseforaclassofautonomousboundaryoperators
relatedto(2.4.1).Letusexplainthesetting.Wespecializetothefinitedimensionalcase
E=CN,N∈N.
LetΩ⊂RnbeadomainwithcompactsmoothboundaryΓ=∂Ω,orΩ=R+n,andlet
p∈(n,∞),µ∈(1/p,1].
Form∈NweconsiderthelinearboundaryoperatorB=(B1,...,Bm),givenby
Bj(x,D)=bjβ(x)trΩDβ,x∈Γ,mj∈{0,...,2m−1},j=1,...,m.
m|≤β|jAsinSection2.1wedefinethecorrespondingnumbersκj∈(0,1)by
κj:=1−mj/2m−1/2mp,j=1,...,m.
Throughoutthissectionitisassumedthat
2mκj−(1−µ+1/p)>(n−1)/p,j=1,...,m.(2.5.1)
ThecoefficientsbjβoftheboundaryoperatorBjaresupposedtosatisfy
bjβ∈B2p,pm(κj−(1−µ+1/p))(Γ,B(CN)),j=1,...,m.(2.5.2)
Dueto(2.5.1)andSobolev’sembeddingitholds
B2p,pm(κj−(1−µ+1/p))(Γ,B(CN))→C(Γ,B(CN)).(2.5.3)
ThusLemma1.3.19andthecontinuitypropertiesofthespatialtracetrΩ(1.3.20)guarantee
thatBmapscontinuously
mB2p,pm(µ−1/p)(Ω,CN)→B2p,pm(κj−(1−µ+1/p))(Γ,CN).
=1jWefurtherassumethatthereisalinearautonomousdifferentialoperatorAoftheform
A(x,D)=aα(x)Dα,x∈Ω,
m=2|α|tsefficiencowithaα∈BUC(Ω,B(CN)),(2.5.4)
suchthat(A,B)satisfiestheellipticityconditions(E)and(LS)fromSection2.1.
SuchasituationarisesinChapter4,where(A,B)isthelinearizationofaquasilinear
problematsomeu0∈B2p,pm(µ−1/p)(Ω,B(CN)).TheretermsoftheformtrΩDβu0,|β|≤mj
enterintothelinearization,whichleadstocoefficientsasin(2.5.2).

2.5ARight-InversefortheBoundaryOperator

99

InProposition4.3.4weusemaximalLp,µ-regularityandtheimplicitfunctiontheoremto
showcontinuousdependenceontheinitialdataforquasilinearproblems.Duetoanonlinear
phasespaceitisthererequiredthatforallµ∈(1/p,1]satisfying(2.5.1)anoperatorB
asabovehasaboundedlinearright-inverseNµ,i.e.,itholdsBNµ=idandNµmaps
continuouslym
B2p,pm(κj−(1−µ+1/p))(Γ,CN)→B2p,pm(µ−1/p)(Ω,CN).
=1jFortheunweightedcase,µ=1,theexistenceofaright-inverseisshownin[65,Proposition
5].TheprooftheremakesuseofthecorrespondingparabolicproblemandthefacttheLp-
spacesonthehalf-lineareinvariantunderrighttranslations.Sincethisisnotthecasefor
theLp,µ-spaces,µ∈(1/p,1),theprooffrom[65]doesnotcarryovertotheweightedcase.
Itisthepurposeofthissectiontoconstructaright-inverseNµalsoforµ∈(1/p,1).The
difficultyisthatforv∈Wp2m(Ω;CN)italwaysholdsBjv∈Wp2mκj(Γ,CN),whichisa
smallerspacethanWp2m(κj−(1−µ+1/p))(Γ,CN).Thustheright-inversecannotdirectlybe
constructedasthesolutionoftheellipticproblem
A(x,D)v=0,x∈Ω,B(x,D)v=(g1,...,gm),x∈Γ,
ifAisrealizedonLp(Ω,CN).Theideaisnowtoshiftthefunctions(g1,...,gm)toahigher
regularityclasswithanappropriateisomorphism,tosolvetheaboveproblemandfinallyto
shiftthesolutionback.Sinceasuitableisomorphismonlyseemstobeavailableonspaces
overRn−1,wehavetolocalizetheproblemoffindingtheright-inversetoRn−1,analogously
totheproofofTheorem2.1.4inthelastsection.
Forclarityreasons,severalassertionsfromthefollowingproofarepostponedinaseriesof
lemmasafterwards.Throughouttherestofthissectionweset
mXµ(Ω):=B2p,pm(µ−1/p)(Ω,CN),Yµ(Γ):=B2p,pm(κj−(1−µ+1/p))(Γ,CN).
=1jProposition2.5.1.LetΩ⊂RnbeadomainwithcompactsmoothboundaryΓ=∂Ω,or
Ω=R+n,letp∈(n,∞)andµ∈(1/p,1]satisfy(2.5.1)andassumethat(A,B)aresubject
to(E),(LS),(2.5.2)and(2.5.4).ThenBhasaboundedlinearright-inverse
mNµ:B2p,pm(κj−(1−µ+1/p))(Γ,CN)→B2p,pm(µ−1/p)(Ω,CN).
=1jProof.(I)Foreachx∗∈Γtheconstantcoefficientoperator(A(x∗,D),B(x∗,D))5is
subjecttothepointwiseconditions(E)and(LS).Weconstructacontinuousright-inverse
Nµ0,x∗:Yµ(Rn−1)→Xµ(R+n)forB(x∗,D)asfollows.Letg=(g1,...,gm)∈Yµ(Rn−1)be
given.Thenwehave
hj:=S−1gj∈Wp2mκj(Rn−1,CN),j=1,...,m,
P5RecallthatBj(x∗,D)=|β|=mjbjβ(x)trΩDβdenotestheprincipalpartofBjforj=1,...,m,and
thatB:=(B1,...,Bm).

100

MaximalLp,µ-RegularityforStaticBoundaryConditions

wheretheoperatorSonLp(Rn−1,CN)isgivenby
S:=(1+(−Δn−1)m)1−µ+1/p.
ItfollowsfromLemma2.2.6thatforallλ>0theuniquesolutionv∈Wp2m(R+n,CN)of
emproblelliptictheλv+A(x∗,D)v=0onR+n,
Bj(x∗,D)v=hjonRn−1,j=1,...,m,
isoftheformv=jm=1Sj(λ)hj,withoperators
Sj(λ)∈BWp2mκj(Rn−1,CN),Wp2m(R+n,CN),j=1,...,m.
WenowdefinetheoperatorNµ0,x∗(λ)by
mNµ0,x∗(λ)g:=SSj(λ)S−1gj.(2.5.5)
=1jHereSactsonthefirstn−1variablesasapointwiserealizationonLp(R+n,CN)=
Lp(R+;Lp(Rn−1,CN)).ItisshowninLemma2.5.4belowthatNµ0,x∗(λ)mapscontinu-
ouslyYµ(Rn−1)→Xµ(R+n).FurtherNµ0,x∗(λ)isinfactaright-inverseforB(x∗,D),since
therealizationofSonLp(R+n,CN)commuteswithB(x∗,D).
(II)Forallx∗∈ΓwechooseaneighbourhoodUx∗⊂Rnofx∗,asmoothdiffeomorphism
ϕx∗:Ux∗→Rnandaradiusrx∗>0with
ϕx∗(x∗)=0,ϕx∗(Ux∗)=B2rx∗(0),ϕx∗(Ux∗∩Ω)⊂R+n,ϕx∗(Ux∗∩Γ)⊂Rn−1.
Forgivenε>0,ifthediameterofUx∗issufficientlysmall,thenbycontinuitythetop
ordercoefficientsofBjsatisfy
sup|bjβ(x∗)−bjβ(x)|<ε,|β|=mj,j=1,...,m.
x∈Γ∩Uex∗
SettingUx∗:=ϕx−∗1Brx∗(0),x∗∈Γ,
weobtainanopencoverx∗∈ΓUx∗forΓ,fromwhichwemaychooseafinitesubcover
partitionofunity{ψi}ofΓ,subordinatetoiUi.
iUicorrespondingtopointsxi∈Γandchartmapsϕi.Therefurtherexistsasmooth
(III)Nowletg=(g1,...,gm)∈Yµ(Γ)begiven.Ifu∈Xµ(Ω)solves
Bu=gonΓ,(2.5.6)
thenforeachithefunctionv=ψiusolves6
Bv=ψig+[B,ψi]uonΓ∩Ui.(2.5.7)
6Recallthat[∙,∙]denotesthecommutatorbracket,i.e.,[B,ψi]u=B(ψiu)−ψiBu.

(2.5.7)

2.5ARight-InversefortheBoundaryOperator

101

DenotingbyΦithepushforwardoperatorcorrespondingtoϕi,i.e.,Φiu=u◦ϕi−1,we
havethatψiusolves(2.5.7)ifandonlyifw=Φi(ψiu)satisfies
BΦiw=Φiψig+[B,ψi]u=:hionRn−1∩Bri(0).(2.5.8)
HereBΦi(x,D):=ΦiB(∙,D)Φi−1(x)denotesthetransformedboundaryoperator,
BΦi(x,D)w=B(∙,D)(w◦ϕi)◦ϕi−1(x),x∈Rn−1∩Bri(0),
andthetransformeddatahi=(h1i,...,him)isidentifiedwithitstrivialextensiontoRn−1
sothatitbelongstoYµ(Rn−1).
AsinStepVoftheproofofTheorem2.1.4weobtainthatthecoefficientsofBΦi,which
aredenotedbybjΦβi,satisfy
bjΦβi∈B2p,pm(κj−(1−µ+1/p))(Rn−1∩Bri(0),CN),|β|≤mj,j=1,...,m.
WedenotebyERn−1∩Bri(0)thecontinuousextensionoperatorfromRn−1∩Bri(0)toRn−1,
givenby(1.3.3),andextendthelowerordercoefficientsofBΦitoRn−1bysetting
bjiβ:=ERn−1∩Bri(0)bjΦβi,|β|<mj,j=1,...,m.
Thetopordercoefficientsareextendedto
bjiβ(x):=bjΦβi(0)+χ(x/2ri)bjΦβi(χ(x/ri)x)−bjΦβi(0),x∈Rn−1,|β|=mj,j=1,...,m,
whereχ∈Cc∞(Rn−1)isanappropriatecut-offfunction.Wedenotetheoperatorwith
extendedcoefficientsbjiβbyBi.Now,ifafunctionw∈Xµ(R+n)solves
Biw=hionRn−1,(2.5.9)
thenw|Rn−1∩Bri(0)solves(2.5.8).
(IV)Tosolve(2.5.9)weconsiderthetoporderconstantcoefficientoperators
Ai,0(D):=ΦiA(∙,D)Φi−1|x=0,Bi,0(D):=Bi(0,D)=bjΦβi(0)trR+nDβ.
|β|=mj
Since(A,B)satisfies(E)and(LS),itfollowsthat(Ai,0,Bi,0)satisfiestheseconditionsas
well.Afunctionwsolves(2.5.9)ifandonlyifitsatisfies
Bi,0(D)w=hi(x)+Bi,sm(x,D)w,x∈Rn−1.(2.5.10)
HeretheoperatorBi,smisgivenby
Bi,sm(x,D)=Bi,0(D)−Bi(x,D),x∈Rn−1,
andthecoefficientsofBi,smaredenotedbybjβi,sm.Byconstructionitholds
bjβi,sm∈B2p,pm(κj−(1−µ+1/p))(Rn−1,CN),|β|≤mj,j=1,...,m,

102

MaximalLp,µ-RegularityforStaticBoundaryConditions

(2.5.12)

andwefurtherhavethatforgivenε
x∈Rsupn−1|bjβi,sm(x)|<ε,|β|=mj,j=1,...,m,
providedthediameteroftheneighbourhoodsUiischosensufficientlysmallfromthebe-
ginning.DuetotheconsiderationsinStepI,acontinuousright-inverseNµ0,i(λ)forBi,0
maybeconstructedasin(2.5.5),forallλ>0.Henceforafunctionw∈Xµ(R+n)tosolve
(2.5.10)itsufficesthatwsatisfies
id−Nµ0,i(λ)Bi,smw=Nµ0,i(λ)hiinR+n.
Lemma2.5.5showsthatifεissufficientlysmallandλissufficientlylargethenthisequation
issolvablebymeansofaNeumannseries,i.e.,
∞Nµi(λ)hi:=Nµ0,i(λ)Bi,smNµ0,i(λ)khi.(2.5.11)
=0kThisyieldsforeachiacontinuoussolutionoperator
Nµi(λ):Yµ(Rn−1)→Xµ(R+n)
for(2.5.9).Thereforeψiusolves(2.5.7)ifψiusatisfies
ψiu=Φi−1Nµi(λ)Φi(ψig+[B,ψi]u)|R+n∩Bri(0).(2.5.12)
(V)Foreachiwechooseafunctionφi∈Cc∞(Ui)with
φi≡1onsuppψi.
Usingthem,wedefinetheoperatorK1(λ):Yµ(Γ)→Xµ(Ω)by
K1(λ)g:=iφiΦi−1(Nµi(λ)Φiψig)|R+n∩Bri(0),g∈Yµ(Γ),
andwefurtherdefinetheoperatorK2:Xµ(Ω)→Xµ(Ω)by
K2(λ)u:=iφiΦi−1(Nµi(λ)Φi[B,ψi]u)|R+n∩Bri(0),u∈Xµ(Ω).
Duetotheconsiderationsinthelaststep,asolutionuof(2.5.6)isasolutionoftheequation
id−K2(λ)u=K1(λ)g,(2.5.13)
whereg∈Yµ(Γ)mustbeappropriatelychosensuchthaterrortermsfromnonempty
intersectionsoftheUicancelwhensummingupinK1andK2.Lemma2.5.6showsthat
thereisaboundedlinearsolutionoperatorQ(λ)for(2.5.13)ifλissufficientlylarge,which
isagainconstructedbymeansofaNeumannseries.
TofindtheappropriategforwhichQ(λ)gsolves(2.5.6),notethatdueto(2.5.13),
(2.5.12)and(2.5.7)itholds
BQ(λ)g=BK1(λ)g+K2(λ)Q(λ)g
=iφiψig+[B,ψi]Q(λ)g−K3(λ)g
=id−K3(λ)g,


2.5ARight-InversefortheBoundaryOperator

103

wherethecorrectionoperatorK3(λ):Yµ(Γ)→Yµ(Γ)comesfromcommutingBwithφiin
i.e.,(2.5.12),K3(λ)h:=i[φi,B]Φi−1Nµi(λ)Φi[ψih+[B,ψi]Q(λ)h]|R+n∩Bri(0),h∈Yµ(Γ).
Lemma2.5.7showsthatforsufficientlylargeλthereisacontinuoussolutionoperator
R(λ):Yµ(Γ)→Xµ(Γ)fortheequation
id−K3(λ)g=g,g∈Yµ(Γ),(2.5.14)
ItthenfollowsthatthecontinuousoperatorNµ:Yµ(Γ)→Xµ(Ω),definedby
Nµg:=Q(λ)R(λ)g,g∈Yµ(Γ),
forsomesufficientlylargeλ,isaright-inverseforB.
Westillhavetoproveseveralassertionsclaimedintheproofabove.
Lemma2.5.2.Letp∈(1,∞),andassumethatα∈[0,1]ands≥0satisfys−2mα≥0.
Thenthepointwiserealizationof(1+(−Δn−1)m)αonLp(R+n,CN)mapscontinuously
Hps(R+n,CN)→Hps−2mα(R+n,CN).(2.5.15)
Restrictingtop∈[2,∞),forσ∈[0,2m(µ−1/p)]thepointwiserealizationoftheoperator
S=(1+(−Δn−1)m)1−µ+1/ponLp(R+n,CN)mapscontinuously
Wp2m−σ(R+n,CN)→B2p,pm(µ−1/p)−σ(R+n,CN).
Proof.Usingextensionsandrestrictions,itsufficestoshowtheassertionforRninstead
ofR+n.Fork∈N0itfollowsfromFubini’stheoremthat
Hpk(Rn,CN)→Hpk(R;Lp(Rn−1,CN))∩Lp(R;Hpk(Rn−1,CN)).(2.5.16)
Theoperators(1−∂y2)k/2and(1+(−Δn−1)m)k/2monLp(R;Lp(Rn−1,CN))withdomains
Hpk(R;Lp(Rn−1,CN))andLp(R;Hpk(Rn−1,CN)),
commuteintheresolventsenseandadmitboundedimaginarypowerswithpoweran-
gleequaltozero,respectively,Therefore,interpolatingtheembedding(2.5.16)with
Lp(R+;Lp(Rn−1,CN))bythecomplexmethod,usingLemmaA.3.4andA.2m)weob-
tainHps(Rn,CN)→Hps(R;Lp(Rn−1,CN))∩Lp(R;Hps(Rn−1,CN)),s>0.
Usingthemixedderivativetheorem(LemmaA.3.3)withtheoperatorsfromabovewethus
havefors≥0that
Hps(Rn,CN)→Hps(1−θ)(R;Hpsθ(Rn−1,CN)),θ∈[0,1].

104

MaximalLp,µ-RegularityforStaticBoundaryConditions

Hencefors≥2mαtheoperator(1+(−Δn−1)m))αmapsHps(Rn,CN)continuouslyinto
Hps(1−θ)(R;Hpsθ−2mα(Rn−1,CN)),θ∈[2mα/s,1].
Nowsupposethats−2mα∈N0.Inthiscase,ifθissuchthatsθ−2mα=k∈N0thenit
alsoholdss(1−θ)=s−2mα−k∈N0.Thus,byFubini’stheorem,
Hps(1−θ)(R;Hpsθ−2mα(Rn−1,CN))=Hps−2mα(Rn,CN),
θ∈[2mα/s,1],sθ−2mα∈N0
whichyieldsthat(2.5.15)holdsfors−2mα∈N0.Thegeneralcasefollowsfromtheinteger
casebycomplexinterpolation.
TheassertedmappingpropertyforSfollowsfrom(2.5.15)byrealinterpolationincase
2m−σ∈/N.For2m−σ∈Nitfollowsfromcomplexinterpolationandtheembedding
Hps(R+n,CN)→Bsp,p(R+n,CN),s≥0,(2.5.17)
whichisvalidforp≥2dueto[82,Theorem2.3.2].
WenextconsiderthemappingpropertiesofanextensionoperatortoR+n.
Lemma2.5.3.Letp∈(1,∞).ConsiderforReλ≥1theoperatorLλ1/2m=(λ+
(−Δn−1)m)1/2m,andthecorrespondingextensionoperatorEλ=e−∙Lλ1/2mfromRn−1to
R+n.ThereisaconstantC>0,whichdoesnotdependonλ,suchthat
1−|EλSh|Lp(R+;Wp2m−mj(Rn−1,CN))≤C|h|B2p,pm(κj−(1−µ+1/p))(Rn−1,CN),(2.5.18)
whereSisdefinedinLemma2.5.2,andfurther
|Eλh|Lp(R+n,CN)≤Cλ−1/2mp|h|Lp(Rn−1,CN).
Proof.(I)Firstobservethatthefunction
ϑ+√z
z→ϑ+(λ+zm)1/2m,z∈Σπ/4m=w∈C\{0}:|argw|<π/4m,
isboundedindependentofReλ≥1and,say,ϑ∈Σ2π/3.Usingthat−Δn−1admitson
Lp(Rn−1,CN)aboundedH∞-calculuswithH∞-angleequaltozero,thisyieldsthatthe
familyratoreopϑ+(−Δn−1)1/2ϑ+(λ+(−Δn−1)m)1/2m−1,Reλ≥1,ϑ∈Σ2π/3,
isuniformlyboundedonLp(Rn−1,CN).Sincefurthertheoperator(−Δn−1)1/2issectorial
withangleofsectorialityequaltozero,itfollowsthattheresolventestimate
|(ϑ+Lλ1/2m)−1|B(Lp(Rn−1,CN))≤C|ϑ|−1,ϑ∈Σ2π/3,
isvalidwithaconstantCindependentofλandϑ.Thisfactimplies
|e−yLλ1/2m|B(Lp(Rn−1,CN))≤C,|Lλe−yL1λ/2m|B(Lp(Rn−1,CN))≤Cy−1,(2.5.19)

2.5ARight-InversefortheBoundaryOperator

105

fory>0,withaconstantCindependentofReλ≥1,sinceforageneratorofananalytic
semigrouptheconstantsintheseestimatesonlydependonthesectorcontainedinthere-
solventsetandontheresolventestimateforthegenerator(seetheproofof[67,Proposition
instance).for(iii)],2.1.1Using(1−Δn−1)s/2asanisomorphismbetweenHps(Rn−1,CN)andLp(Rn−1,CN))that
commuteswithLλ1/2m,weobtainfrominterpolationthat(2.5.19)remainsvalidifone
replacesB(Lp(Rn−1,CN))byB(Wps(Rn−1,CN))fors≥0.
(II)Inthisstepwefollowtheproofof[68,Proposition6.2].Takev∈Wp2mκj(Rn−1,CN)
andletv=a+bwitha∈Wp2m−mj−1(Rn−1,CN)andb∈Wp2m−mj(Rn−1,CN).Usingthat
theoperator(1−Δn−1)1/2Lλ−1/2misuniformlyboundedinReλ≥1,andusingfurther
(2.5.19),weobtainfory>0that
|e−yLλv|Wp2m−mj(Rn−1,CN)≤C|Lλe−yLλa|Wp2m−mj−1(Rn−1,CN)+|e−yLλb|Wp2m−mj(Rn−1,CN)
1−≤Cy|a|Wp2m−mj−1(Rn−1,CN)+C|b|Wp2m−mj(Rn−1,CN).
Takingtheinfimumoveraandbontheright-handsideleadsto
|e−yLλv|Wp2m−mj(Rn−1,CN)≤Cy−1Ky,v,Wp2m−mj−1(Rn−1,CN),Wp2m−mj(Rn−1,CN)
fory>0,whereKdenotestheK-functionalfromrealinterpolationtheory(see[68]).It
nowfollowsfromthedefinitionoftherealinterpolationfunctor(∙,∙)1−1/p,pthat
|e−∙Lλv|Lp(R+,Wp2m−mj(Rn−1,CN))≤C|v|B2p,pmκj(Rn−1,CN),
withaconstantCindependentofλ.Theestimate(2.5.18)isnowaconsequenceofthe
factthatS=(1+(−Δn−1)m)1−µ+1/pisanisomorphismbetweenB2p,pmκj(Rn−1,CN)and
B2p,pm(κj−(1−µ+1/p))(Rn−1,CN).
(III)Thefunctionz→expy(λ1/2m−(λ+zm)1/2m)isholomorphicandboundedon
Σπ/4m,independentofy>0andReλ≥1.UsingagaintheboundedH∞-calculusof
−Δn−1weobtainthatthereisaconstantC>0,independentofyandλ,suchthat
|e−yLλ1/2mh|Lp(Rn−1,CN)≤Ce−yλ1/2m|h|Lp(Rn−1,CN),h∈Lp(Rn−1,CN).
TakingtheLp(R+)normwithrespecttoyshowsthesecondassertedestimate.
Independenceonλweconsiderthecontinuitypropertiesoftheright-inversesNµ0,x∗(λ)on
thehalf-space,definedin(2.5.5).
Lemma2.5.4.InthesettingoftheproofofProposition2.5.1,considerforx∗∈Γthe
ratoreopmNµ0,x∗(λ)g:=SSj(λ)S−1gj,g=(g1,...,gm)∈Yµ(Rn−1).
=1jThenforσ1∈[0,µ−1/p]wehave
|Nµ(λ)g|B2p,pm(µ−1/p−σ1)(R+n,CN)λ1j=1max,...,m|gj|B2p,pm(κj−(1−µ+1/p))(Rn−1,CN)(2.5.20)
0,x∗−σ
m+λ−(1−µ+1/p)−1/2mpλ1−2mj|gj|Lp(Rn−1,CN),

106

MaximalLp,µ-RegularityforStaticBoundaryConditions

and,forσ2∈[0,2m−1/p),
mj|Nµ0,x∗(λ)g|Bp,11/p+σ2(R+n,CN)λ−(1−σ2/2m)j=1max,...,mλ1−2m|gj|Lp(Rn−1,CN)(2.5.21)
+λ1−µ+1/p+1/2mp|gj|B2p,pm(κj−(1−µ+1/p))(Rn−1,CN)
Moreover,forσ3∈[0,κj−(1−µ+1/p))andβ∈N0nwith|β|≤mjitholds
|β−|mj|trR+nDβNµ0,x∗(λ)g|B2p,pm(κj−(1−µ+1/p)−σ3)(Rn−1,CN)λ−σ3λ−2ml=1max,...,m(2.5.22)
|gl|B2m(κl−(1−µ+1/p))(Rn−1,CN)+λ−(1−µ+1/p)−1/2mpλ1−2mml|gl|Lp(Rn−1,CN),
p,pfurtherand|trR+nDβNµ0,x∗(λ)g|Lp(Rn−1,CN)λ−1−2|βm|maxλ1−2mml|gl|Lp(Rn−1,CN)(2.5.23)
,...,m=1l+λ1−µ+1/pλ1/2mp|gl|B2p,pm(κl−(1−µ+1/p))(Rn−1,CN).
Proof.(I)DuetoLemma2.2.6,forj=1,...,mtheoperatorSj(λ)isoftheform
Sj(λ)=Tj(λ)Lλ1−mj/2mEλ,
whereTj(λ)∈BLp(R+n,CN),Wp2m(R+n,CN),Lλ=λ+(−Δn−1)mandEλ=e−∙Lλ1/2m.
m1−mj/2m
Usingthatthefunctionz→λ1−(mλj+/2zm)+z(2m−mj)/2isuniformlyboundedfor,say,z∈Σπ/4m,
andthepropertiesofTj(λ)statedinLemma2.2.6wemayrewriteSj(λ)to
Sj(λ)=Tj(λ)(−Δn−1)2m2−mj+λ1−2mmjEλ,
whereTj(λ)hasforλ>0andj=1,...,mtheproperty
|Tj(λ)v|B2p,qm−s(R+n,CN)λ−s/2m|v|Lp(R+n,CN),s∈[0,2m],q∈[1,∞].(2.5.24)
Theproofof[25,Lemma4.3]showsthatTj(λ)isaconvolutionoperatorwithrespectto
x∈Rn−1.Therefore−Δn−1commuteswithTj(λ),andthusalsowithTj(λ).Nowitfollows
from[7,LemmaIII.4.9.2]thatScommuteswithTj(λ).Togetherwith(2.5.24)weobtain
|STj(λ)v|Wp2m(µ−1/p−σ1)(R+n,CN)λ−σ1λ−(1−µ+1/p)|Sv|Lp(R+n,CN).(2.5.25)
(II)Toshow(2.5.20),firstobservethat
|Nµ0,x∗(λ)g|B2p,pm(µ−1/p−σ1)(R+n,CN)
2m−mj−1
j=1max,...m|STj(λ)(−Δn−1)2EλSgj|B2p,pm(µ−1/p−σ1)(R+n,CN)
m++j=1max,...,m|STj(λ)λ1−2mjEλS−1gj|B2p,pm(µ−1/p−σ1)(Rn,CN).
Henceforeachjwehavetoestimatethesetwosummands.Forthefirstsummandweuse
(2.5.24)andtheLemmas2.5.2and2.5.3toobtain
m−m2j|STj(λ)(−Δn−1)2EλS−1gj|B2p,pm(µ−1/p−σ1)(R+n,CN)λ−σ1|gj|B2p,pm(κj−(1−µ+1/p))(Rn−1,CN).

2.5ARight-InversefortheBoundaryOperator

107

Usinginaddition(2.5.25),wehaveforthesecondsummand
|STj(λ)λ2mEλSgj|B2p,pm(µ−1/p−σ1)(Rn,CN)
+1−mj−1
mjλ−σ1λ−(1−µ+1/p)−1/2mpλ1−2m|gj|L(Rn−1,CN).
p(III)For(2.5.21)wefirstestimate,asinthelaststep,
|Nµ(λ)g|B1/p+σ2(Rn,CN)j=1max,...,m|STj(λ)(−Δn−1)2EλSgj|B1/p+σ2(Rn,CN)
p,1+p,1+
0,x∗2m−mj−1
+j=1max,...,m|STj(λ)λ2mEλSgj|B1/p+σ2(Rn,CN).
+1p,1−mj−1
Using(2.5.24),weobtainforthefirstsummand
2m−mj−1
+1p,|STj(λ)(−Δn−1)2EλSgj|B1/p+σ2(Rn,CN)
λ−(1−σ2/2m)λ1−µ+1/p+1/2mp|gj|2m(κj−(1−µ+1/p)),
Bp,p(Rn−1,CN)
summandsecondtheforandmmjjBp,1(R+,C)
|STj(λ)λ1−2mEλS−1gj|1/p+σ2nNλ−(1−σ2/2m)λ1−2m|gj|Lp(Rn−1,CN).
(IV)For(2.5.22)weestimate
β0,x∗0,x∗
|trR+nDNµ(λ)g|B2p,pm(κj−(1−µ+1/p)−σ3)(Rn−1,CN)|Nµ(λ)g|2m(µ−1/p−σ3−mj2m−|β|)nN,
Bp,p(R+,C)
andthus(2.5.22)followsfrom(2.5.20).In(2.5.23),thetraceoperatormeetstheLp-norm.
ForthisweusethattrR+niscontinuous
B1/p(R+n,CN)→Lp(Rn−1,CN),
1p,Then2.9.3].Theorem[82,see,x0,x0β|trR+nDNµ∗(λ)g|Lp(Rn−1,CN)|Nµ∗(λ)g|B1/p+|β|(Rn,CN),
+1p,and(2.5.23)followsfrom(2.5.21).
WenextprovetheconvergenceoftheNeumannseriesin(2.5.11).
Lemma2.5.5.InthesettingoftheproofofProposition2.5.1,foreachitheseries
∞Nµi(λ)=Nµ0,i(λ)Bi,smNµ0,i(λ)k
=0kexistsinBYµ(Rn−1),B2p,pm(µ−1/p−σ1)(R+n,CN),providedεissufficientlysmallandλis
sufficientlylarge.Forσ1∈[0,µ−1/p]itholds
σ−i|Nµ(λ)g|B2m(µ−1/p−σ1)(Rn,CN)λ1|g|Yµ(Rn−1)(2.5.26)
p,p+mmj+λ−(1−µ+1/p)−1/2mpλ1−2m|gj|Lp(Rn−1,CN).
=1j

108MaximalLp,µ-RegularityforStaticBoundaryConditions
Moreover,forσ2∈[0,2m−1/p)wehave
|Nµi(λ)g|B1/p+σ2(Rn,CN)≤Cλ−(1−σ2/2m)λ1−µ+1/p+1/2mp|g|Yµ(Rn−1)(2.5.27)
+1p,m1−mj
+λ2m|gj|Lp(Rn−1,CN).
=1jProof.(I)Itfollowsfrom(2.5.20)thatfork∈N0itholds
0,ii,sm0,ik
|Nµ(λ)(BNµ(λ))g|B2p,pm(µ−1/p−σ1)(R+n,CN)(2.5.28)
λ1j=1max,...,m|((BNµ(λ))g)j|B2p,pm(κj−(1−µ+1/p))(Rn−1,CN)
−σi,sm0,ik
mj+λ−(1−µ+1/p)−1/2mpλ1−2m|((Bi,smNµ0,i(λ))kg)j|Lp(Rn−1,CN),
andfurther,dueto(2.5.21),
0,ii,sm0,ik
|Nµ(λ)(BNµ(λ))g|Bp,11/p+σ2(R+n,CN)(2.5.29)
λ−(1−σ2/2m)maxλ1−µ+1/p+1/2mp|((Bi,smNµ0,i(λ))kg)j|2m(κj−(1−µ+1/p))n−1N
j=1,...,mBp,p(R,C)
mj+λ1−2m|((Bi,smNµ0,i(λ))kg)j|Lp(Rn−1,CN).
(II)Foreachjweconsiderthesummandsin(2.5.28).Forthefirstsummandwehave
,i0i,sm|((BNµ(λ)g)j|B2p,pm(κj−(1−µ+1/p))(Rn−1,CN)(2.5.30)
,i0βi,smmax|bjβtrR+nDNµ(λ)g|B2m(κj−(1−µ+1/p))(Rn−1,CN).
|β|≤mjp,p
ApplyingLemma1.3.19andusing(2.5.3)foreachβweobtainforsmallδ>0that
i,sm|bjβtrR+nDβNµ0,i(λ)g|B2m(κj−(1−µ+1/p))(Rn−1,CN)
p,p,i0βi,sm≤C|bjβ|L∞(Rn−1,CN)|trR+nDNµ(λ)g|B2p,pm(κj−(1−µ+1/p))(Rn−1,CN)
,i0β+Cε|trR+nDNµ(λ)g|B2p,pm(κj−(1−µ+1/p)−δ)(Rn−1,CN)
=:I1+I2.
yields)(2.5.22estimateTheI1≤(Cε+Cελ)l=1max,...,m|gl|B2p,pm(κl−(1−µ+1/p))(Rn−1,CN)
−1/2m
ml+λ−(1−µ+1/p)−1/2mpλ1−2m|gl|Lp(Rn−1,CN),
whereCisindependentofεandCεisindependentofλ.Inthesamewayweobtain
δ−I2≤Cελl=1max,...,m|gl|B2p,pm(κj−(1−µ+1/p))(Rn−1,CN)
m+λ−(1−µ+1/p)−1/2mpλ1−2ml|gl|Lp(Rn−1,CN).

2.5ARight-InversefortheBoundaryOperator

109

Combiningtheseestimateswith(2.5.30)leadsto
|(Bi,smNµ0,i(λ)g)j|B2p,pm(κj−(1−µ+1/p))(Rn−1,CN)
≤(Cεi+Cεiλ−δ)l=1max,...,m|gl|Wp2m(κl−(1−µ+1/p))(Rn−1,CN)
+λ−(1−µ+1/p)λ−1/2mpλ1−2mml|gl|Lp(Rn−1,CN).
Notethatforg=(Bi,smNµ0,i(λ))k−1gtheright-handsideaboveisofthesametypeasthe
right-handsidein(2.5.28)withk−1insteadofkandtheadditionalfactor(Cε+Cελ−δ).
Forthesecondsummandin(2.5.28)wehave,using(2.5.23),
|β||(Bi,smNµ0,i(λ)g)j|Lp(Rn−1,CN)≤C|bjβi,sm|L∞(Rn−1,CN)λ−1−2m
mll=1max,...,mλ1−µ+1/p+1/2mp|gl|B2p,pm(κl−(1−µ+1/p))(Rn−1,CN)+λ1−2m|gl|Lp(Rn−1,CN),
yieldshwhicλ−(1−µ+1/p)λ−1/2mpλ1−2mmj|((Bi,smNµ0,i(λ))kg)j|Lp(Rn−1,CN)
≤(Cε+Cελ−1/2m)l=1max,...,m|((Bi,smNµ0,i(λ))k−1g)l|B2p,pm(κl−(1−µ+1/p))(Rn−1,CN)
+λ−(1−µ+1/p)−1/2mpλ1−2mml|((Bi,smNµ0,i(λ)g)k−1)l|Lp(Rn−1,CN).
Again,theright-handsideisofthesametypeastheright-handsidein(2.5.28),withk
replacedbyk−1andtheadditionalfactorCε+Cελ−1/2m.
(III)Iteratingtheaboveestimatesyieldsfork∈N0that
|Nµ0,i(λ)(Bi,smNµ0,i(λ))kg|B2p,pm(µ−1/p−σ1)(R+n,CN)(Cε+Cελ−τ)k
mλ−σ1|g|Yµ(Rn−1)+λ−(1−µ+1/p)−1/2mpλ1−2mmj|gj|Lp(Rn−1,CN),
=1jwithsomeτ>0.ThisimpliesthatNµi(λ)existsinB(Yµ(Rn−1),B2p,pm(µ−1/p−σ1)(R+n,CN))
andadmitstheassertedestimate(2.5.26),providedwefirstchooseεsufficientlysmalland
thenλsufficientlylarge.
(IV)StartinginStepIIwith(2.5.29)insteadof(2.5.28)oneobtains(2.5.27)inasimilar
fashion,usingtheestimatesfromLemma2.5.4.
Thenextlemmashowstheuniquesolvabilityequation(2.5.13).
Lemma2.5.6.InthesettingofProposition2.5.1,considertheoperators
K1(λ)g=iφiΦi−1(Nµi(λ)Φiψig)|R+n∩Bri(0),g∈Yµ(Γ),
K2(λ)u=iφiΦi−1(Nµi(λ)Φi[B,ψi]u)|R+n∩Bri(0),u∈Xµ(Ω).
Foreachg∈Yµ(Γ)theequationid−K2(λ)u=K1(λ)ghasauniquesolutionu:=
Q(λ)g∈Xµ(Ω),providedλissufficientlylarge.Forσ1∈[0,µ−1/p]thesolutionoperator

110

satisfies

MaximalLp,µ-RegularityforStaticBoundaryConditions

(2.5.31)

|Q(λ)g|B2p,pm(µ−1/p−σ1)(Ω,CN)(2.5.31)
λ−σ1|g|Y(Γ)+λ−(1−µ+1/p)−1/2mpλ1−2m|gj|L(Γ,CN),
mmj
pµ=1jandforσ2∈[0,2m−1/p)wefurtherhave
|Q(λ)g|B1/p+σ2(Ω,CN)(2.5.32)
1p,λ−(1−σ2/2m)λ1−µ+1/p+1/2mp|g|Yµ(Γ)+λ1−2m|gj|Lp(Γ,CN).
mmj
=1jProof.(I)Weconcentrateon(2.5.31),similarargumentsleadto(2.5.32).Weintendto
showtheabsoluteconvergenceoftheNeumannseries
∞Q(λ):=K2(λ)kK1(λ)
=0kinBYµ(Γ),B2p,pm(µ−1/p−σ1)(Ω,CN).Itfollowsfrom(2.5.26)that
|K2(λ)u|B2p,pm(µ−1/p−σ1)(Ω,CN)≤max|NiΦi[B,ψi]u|B2p,pm(µ−1/p−σ1)(Rn,CN)
i+λ−σ1max|Φi[B,ψi]u|Yµ(Rn−1)
immj+λ−(1−µ−1/p)−1/2mpλ1−2m|Φi[Bj,ψi]u|Lp(Rn−1,CN).
=1jIfmj=0forsomejthen[Bj,ψi]=0,thusweassumethatmj≥1foreachj=1,...,m
inthesequel.As[Bj,ψi]isoforderatmostmj−1wehaveforeachithat
|Φi[B,ψi]u|Yµ(Rn−1)|[B,ψi]u|Yµ(Γ∩Ui)|u|B2m(µ−1/p)−1(Ω,CN),
p,pandfurther,foreachj=1,...,m,
|Φi[Bj,ψi]u|Lp(Rn−1,CN)|u|B1/p+mj−1(Ω,CN).
1p,yieldsThisσ−1|K2(λ)u|B2p,pm(µ−1/p−σ1)(Ω,CN)λ|u|B2p,pm(µ−1/p)−1(Ω,CN)(2.5.33)
mmj+λ−(1−µ−1/p)−1/2mpλ1−2m|u|1/p+m−1.
Bp,1j(Ω,CN)
=1jimplies(2.5.27)er,vMoreomj|K2(λ)u|1/p+mj−1λ−1/2mλ−(1−2m)λ1−µ+1/p+1/2mp|u|2m(µ−1/p)−1N
Bp,1(Ω,CN)Bp,p(Ω,C)
m+λ2m|u|B1/p+ml−1(Ω,CN).
1−ml
1p,=1l

2.5ARight-InversefortheBoundaryOperator

111

Iteratingtheaboveestimatesweobtainfork∈N
|K2(λ)u|B2m(µ−1/p−σ1)(Ω,CN)(Cλ)2mλ1|u|B2p,pm(µ−1/p)−1(Ω,CN)(2.5.34)
k−k−1−σ
p,pmmj+λ−(1−µ−1/p)−1/2mpλ1−2m|u|1/p+mj−1,
Bp,1(Ω,CN)
=1j

wheretheconstantCisindependentofλ.
(II)WenowestimateK1(λ).From(2.5.26)weinfer
|K1(λ)g|B2p,pm(µ−1/p)−1(Ω,CN)imax|NiΦiψig|B2p,pm(µ−1/p)−1(R+n,CN)
mmλ−1/2m|g|Yµ(Γ)+λ−(1−µ+1/p)−1/2mpλ1−2mj|gj|Lp(Γ,CN),
=1j(2.5.27),fromfurther,andmj|K1(λ)g|Bp,11/p+mj−1(Ω,CN)λ−1/2mλ−1−2mλ1−µ+1/p+1/2mp|g|Yµ(Γ)
m+λ1−2mml|gl|Lp(Γ,CN)
=1lforj=1,...,m.Usingtheseestimatesforu=K1(λ)gin(2.5.34)weobtainfork∈N0that
|K2(λ)kK1(λ)g|B2m(µ−1/p−σ1)(Ω,CN)≤(Cλ)−k/2mλ−σ1|g|Yµ(Γ)
p,pm+λ−(1−µ+1/p)−1/2mpλ1−2mmj|gj|Lp(Γ,CN).
=1jThisyieldstheabsoluteconvergenceoftheNeumannseriesandtheestimateforQ(λ)as
asserted,providedλissufficientlylarge.
Lastbutnotleastweconsidertheequation(2.5.14).

Lemma2.5.7.InthesettingofProposition2.5.1,considertheoperator
K3(λ)h=i[φi,B]Φi−1Nµi(λ)Φi[ψih+[B,ψi]Q(λ)h]|R+n∩Bri(0),h∈Yµ(Γ).
Ifλissufficientlylargethenforeachg∈Yµ(Γ)thereisauniquesolutionh=R(λ)g∈
Yµ(Γ)of(id−K3)h=g.ThesolutionoperatorR(λ)iscontinuousonYµ(Γ).
Proof.WeshowtheabsoluteconvergenceoftheNeumannseriesk∞=0K3(λ)kinB(Yµ(Γ)).
Weassumethatmj≥1forj=1,...,minthesequel,otherwisethecorrespondingcom-
mutatorsvanish.Using(2.5.26)andthatthecommutators[φi,Bj]areoflowerorderwe

112

MaximalLp,µ-RegularityforStaticBoundaryConditions

obtain|K3(λ)g|Yµ(Γ)imax|Nµi(λ)Φi[ψig+[B,ψi]Q(λ)g]|B2p,pm(µ−1/p)−1(R+n,CN)
λ−1/2mmax|Φi[ψig+[B,ψi]Q(λ)g]|Yµ(Rn−1)
im+λ−(1−µ+1/p)−1/2mpλ1−2mmj|Φi[ψigj+[Bj,ψi]Q(λ)g]|Lp(Rn−1,CN)
=1jλ−1/2mimax|g|Yµ(Γ)+|[B,ψi]Q(λ)g|Yµ(Γ)
mm+λ−(1−µ+1/p)−1/2mpλ1−2mj(|gj|Lp(Γ,CN)+|[Bj,ψi]Q(λ)g|Lp(Γ,CN)).
=1jWefurtherinferfrom(2.5.31)that
mm|[B,ψi]Qg|Yµ(Γ)λ−1/2m|g|Yµ(Γ)+λ−(1−µ+1/p)−1/2mpλ1−2mj|gj|Lp(Γ,CN),
=1jthat(2.5.32)fromand|[Bj,ψi]Q(λ)g|Lp(Γ,CN)|Q(λ)g|Bp,11/p+mj−1(Ω,CN)
mλ−1/2mλ−(1−2mmj)λ1−µ+1/p+1/2mp|g|Yµ(Γ)+λ1−2mml|gl|Lp(Γ,CN),
=1lyieldshwhicm|K3(λ)g|Yµ(Γ)λ−1/2m|g|Yµ(Γ)+λ−(1−µ+1/p)−1/2mpλ1−2mml|gl|Lp(Γ,CN).
=1lUsing(2.5.27),wealsoobtainforl=1,...,mthat
|(K3(λ)g)l|Lp(Γ,CN)
λ−1/2mλ−1−2mmlλ1−µ+1/p+1/2mp|g|Yµ(Γ)+|[B,ψi]Q(λ)g|Yµ(Γ)
+λ1−2m|gj|Lp(Γ,CN)+|[Bj,ψi]Q(λ)g|Lp(Γ,CN).
mmj
=1jHencewehavefork∈N0,withaconstantCthatisindependentofλ,
m|K3(λ)kg|Yµ(Γ)≤(Cλ)−k/2m|g|Yµ(Γ)+λ−(1−µ+1/p)−1/2mpλ1−2mml|gl|Lp(Γ,CN).
=1lThisyieldstheconvergenceofk∞=0K3(λ)kinB(Yµ(Γ))andthecontinuityofthesolution
operatorR(λ),providedλissufficientlylarge.

3Chapter

MaximalLp,µ-Regularityfor
BoundaryConditionsofRelaxation
ypTe

InthischapterweshowmaximalLp,µ-regularityforvector-valuedparabolicinitial-
boundaryvalueproblemsofrelaxationtype,generalizingtheresultsofbyDenk,Prüss,&
Zacher[26].TheapproachisanalogoustothatinChapter2forthecaseofstaticbound-
aryconditions.Thussometimeswearebrief,butalsorepeatsomeargumentsfromthe
lastchapterfortransparency.Wefirstdescribetheapproachandtheinvolvedanisotropic
functionspacesindetail,andthenprovethemainresult,Theorem3.1.4,bysolvingthe
half-spaceproblemandperformingaperturbationandlocalizationprocedure.Forthege-
ometryoftheboundaryofadomainanddifferentialoperatorsdefinedonthemwereferto
theAppendeciesA.1andA.5.

3.1TheProblemandtheApproachinWeightedSpaces
ProblemTheFortheunknownvector-valuedfunctions

u=u(t,x)∈E,ρ=ρ(t,x)∈F,
weconsiderlinearinhomogeneous,non-autonomous,parabolicinitial-boundaryvalueprob-
lemsofrelaxationtype,i.e.,

∂tu+A(t,x,D)u=f(t,x),x∈Ω,t∈J,
∂tρ+B0(t,x,D)u+C0(t,x,DΓ)ρ=g0(t,x),x∈Γ,t∈J,
Bj(t,x,D)u+Cj(t,x,DΓ)ρ=gj(t,x),x∈Γ,t∈J,j=1,...,m,(3.1.1)
u(0,x)=u0(x),x∈Ω,
ρ(0,x)=ρ0(x),x∈Γ.

114

MaximalLp,µ-RegularityforBoundaryConditionsofRelaxationType

HereΩ⊂RnisassumedtobeadomainwithcompactsmoothboundaryΓ=∂Ω,J=
(0,T)isafiniteinterval,T>0,andE,FareBanachspacesofclassHT.Theunknownu
livesonJ×Ω,whiletheunknownρlivesonJ×Γ,i.e.,itisonlypresentontheboundary
Γ.Itisassumedthatthedynamicequationforuandthestaticboundaryequationstake
placeinE,andthatthedynamicequationforρtakesplaceinF.Consequently,theright-
handsidesf,g1,...,gm,andtheinitialvalueu0takevaluesinE,whileg0andρ0take
.FinaluesvFormallyoneobtainstheproblem(2.1.1)withstaticboundaryconditionsbysettingρ≡0
anddroppingtheseconddynamicequation.
ThedifferentialoperatorAisgivenby
A(t,x,D)=aα(t,x)Dα,x∈Ω,t∈J,
m2|≤α|wherem∈NandD=−i,withtheeuclidiangradient=(∂x1,...,∂xn)onRn,and
coefficientsaα(t,x)∈B(E).HencetheorderofAis2m.TheboundaryoperatorsBjare
formtheofBj(t,x,D)=bjβ(t,x)trΩDβ,x∈Γ,t∈J,j=0,...,m,
m|≤β|jwheremj∈{0,...,2m−1}istheorderofBj,andthecoefficientssatisfy
b0β(t,x)∈B(E,F),bjβ(t,x)∈B(E),j=1,...,m.
ObservethatB=(B0,...,Bm)onlyactsonu,inawaythatfirsttheeuclidianderivatives,
andthenthespatialtracetrΩisapplied.
TheoperatorsC=(C0,...,Cm)onlyactonρ,inthefollowingway.For(almostevery)t∈J
itisassumedthatCj(t,∙,DΓ)isalinearmap
C∞(Γ;F)→L1(Γ;F),
suchthatforallj=0,...,m,alllocalcoordinatesgforΓandallρ∈C∞(Γ;F)itholds
Cj(t,∙,DΓ)ρ◦g(x)=cjgγ(t,x)Dnγ−1(ρ◦g)(x),x∈g−1(Γ∩U),t∈J,
k|≤γ|jwhereU⊂Rnisthedomainofthechartcorrespondingtog.HerewehaveDn−1=−in−1,
withtheeuclidiangradientn−1=(∂x1,...,∂xn−1)onRn−1,andkj∈N0istheorderof
Cj.Thelocalcoefficientscjgγ,thatmaydependonthecoordinatesg,areassumedtosatisfy
c0gγ(t,x)∈B(F),cjgγ(t,x)∈B(F,E),j=1,...,m.
WedonotassumethatanoperatorCjhasglobalcoefficients,inthesensethatthereare
functionscjγonΓsatisfyingcjgγ=cjγ◦ginallcoordinatesg.Incontrasttothat,the
coefficientsofBaregloballydefinedonΓ.WewriteCj(DΓ)withDΓ=−iΓ,sinceforCj

3.1TheProblemandtheApproachinWeightedSpaces

115

wethinkofanoperatorintermsofthesurfacegradientΓ.WerefertoAppendixA.5for
moreinformationsonthesurfacegradientandgeneraldifferentialoperatorsonaboundary
actingonvector-valuedfunctions.
Finally,itisassumedthateachoftheoperatorsBjandatleastoneoperatorCjare
nontrivial.IfanoperatorCjistrivial,i.e.,Cj≡0,thenwesetkj:=−∞foritsorder.
Weconsiderthreeproblemsthatfitintotheaboveframework.Furtherexamplesarelisted
3].Section[26,inExample3.1.1.Alinearizedreaction-diffusionsystemwithsurfacediffusion,

∂tu−Δu=f(t,x),x∈Ω,t>0,
∂tu+∂νu−ΔΓu=g(t,x),x∈Γ,t>0,(3.1.2)
u(0,x)=u0(x),x∈Ω,
where∂ν=ν(x)∙trΩΩdenotestheouternormalderivativeand−ΔΓistheLaplace-
BeltramioperatoronΓ.Thelatterisinlocalcoordinatesggivenby
1−n|G|k,l=1
ΔΓρ◦g=1∂xk|G|gkl∂xl(ρ◦g),
whereGisthefirstfundamentalformcorrespondingtogandgklarethecomponentsof
G−1,cf.AppendixA.1.Theproblem(3.1.2)canbecastintheform(3.1.1)asfollows,

∂tu−Δu=f(t,x),x∈Ω,t>0,
∂tρ+∂νu−ΔΓρ=g(t,x),x∈Γ,t>0,
trΩu−ρ=0,x∈Γ,t>0,
u(0,x)=u0(x),x∈Ω,
ρ(0,x)=trΩu0(x),x∈Γ.
HencetheunknownρissimplythetraceofuonΓ.TheoperatorA(D)=−Δisoforder
2,thusm=1.WefurtherhaveB0(x,D)=∂ν,C0(x,DΓ)=−ΔΓ,B1=trΩ,C1=−id,
suchthatm0=1,k0=2,andm1=k1=0.

NeglectingtheLaplace-Beltramioperator,weobtain

∂tu−Δu=f(t,x),x∈Ω,t>0,
∂tu+∂νu=g(t,x),x∈Γ,t>0,(3.1.3)
u(0,x)=u0(x),x∈Ω.
Asabove,thisproblemcanbecastintheform(3.1.1)bytakingtrΩu=ρasstaticboundary
condition.Theonlydifferenceto(3.1.2)isthatC0≡0,hencek0=−∞.

116

MaximalLp,µ-RegularityforBoundaryConditionsofRelaxationType

(3.1.4)

TransformingtheStefanproblemwithsurfacetensiontoafixeddomain,thelinearization
oftheresultingproblemisoftheform
∂tu−Δu=f(t,x),x∈Ω,t>0,
∂tρ+∂νu=g0(t,x),x∈Γ,t>0,
u+ΔΓρ=g1(t,x),x∈Γ,t>0,(3.1.4)
u(0,x)=u0(x),x∈Ω,
ρ(0,x)=ρ0(x),x∈Γ.
Herethegraphofρ(t,∙)overΓdescribesthefreeboundaryattimet.ThemaximalLp-
regularityfor(3.1.4)isthebasictoolin[37]toshowanalyticityofthefreeboundary.
Thisproblemstructurallydiffersfrom(3.1.2)and(3.1.3),sinceρisnotsimplythetrace
ofu,andthestaticcouplingoftheseunknownsisnontrivial.ItholdsB1=trΩ,m1=0,
C1(x,DΓ)=ΔΓ,k1=2,andfurtherm0=1andk0=−∞.

TheApproachintheLp,µ-Spaces
ThemaximalLp,µ-regularityapproachfor(3.1.1)isasfollows.Let
p∈(1,∞),µ∈(1/p,1].
Welookforsolutions(u,ρ)sothatthefirstcomponentusatisfies
u∈Eu,µ:=W1p,µ(J;Lp(Ω;E))∩Lp,µ(J;Wp2m(Ω;E)).
Asinthestaticcase,theresultsofSection1.3showthatthisregularityassumptiononu
impliesnecessarilyf∈E0,µ:=Lp,µ(J;Lp(Ω;E)),u0∈Xu,µ:=B2p,pm(µ−1/p)(Ω;E),
thatfurtherandg0∈F0,µ:=Wκp,µ0(J;Lp(Γ;F))∩Lp,µ(J;Wp2mκ0(Γ;F)),
andgj∈Fj,µ:=Wκp,µj(J;Lp(Γ;E))∩Lp,µ(J;Wp2mκj(Γ;E)),j=1,...,m,
wherewesetmj1
κj:=1−2m−2mp,j=0,...,m.
Forconveniencewewrite
Fµ:=F0,µ×...×Fm,µ,g=(g0,...,gm)∈Fµ.
Therefore,in(3.1.1)thedynamicequationforutakesplaceinE0,µ,thedynamicequation
forρinF0,µ,andthestaticboundaryconditionsinFj,µ,j=1,...,m,respectively.

3.1TheProblemandtheApproachinWeightedSpaces

117

Lookingforoptimalregularity,thespaceEρ,µforρshouldnowbesuchthat,assuming
smoothnessofthecoefficients,allthesummandsinthetermsin(3.1.1)whereρisinvolved
belongtothespacewheretherespectiveequationtakesplace.Itcanbeseenasin[26,
that2]SectionEρ,µ=Wp,µ1+κ0J;Lp(Γ;F)∩Lp,µJ;Wpl+2mκ0(Γ;F)
∩W1p,µJ;Wp2mκ0(Γ;F)∩Wκp,µjJ;Wpkj(Γ;F)
eJ∈jsatisfiestheserequirements.Herewehaveusedtheabbreviations
J:=j∈{0,...,m}:kj=−∞,lj:=kj−mj+m0,l:=j=0max,...mlj.
ObservethatJjustcollectstheindicesjforwhichanoperatorCjisnontrivial,andthat
withtheabovenotationsitholds
kj+2mκj=lj+2mκ0≤l+2mκ0.
Proposition1.3.2showsthatthereisredundancyintheabovedefinitionofEρ,µ,depending
ontherelationofland2m.TherearethreepossiblequalitativeshapesoftheNewton
polygonassociatedtoEρ,µ(seeSection1.3.1).Thepoints(0,1+κ0)and(l+2mκ0,0)
arealwaysverticesoftheNewtonpolygon.Thelinethroughthepoints(0,1+κ0)and
(2mκ0,1)intersects(2m+2mκ0,0),sothatW1p,µ(J;Wp2mκ0(Γ;F))isredundantforl≥
2m,andthepoints(kj,κj)determinetheremainingverticesofthenontrivialpartofthe
polygon.Moreover,thelinesthroughthepoints(kj,κj)and(kj+2mκj,0)areparallelfor
j=0,...,m.Thusforl≤2mthespacescorrespondingtothepoints(kj,κj)areredundant,
and(2mκ0,1)isavertexifl<2m.Belowwegivetheprecisenonredundantdescription
.Eofρ,µIneachcase,Theorem1.3.6yieldsthetemporaltracespaceofρatt=0,whichisdenoted
ybXρ,µ:=trt=0Eρ,µ,
and,ifitexists,of∂tρatt=0,whichisdenotedby
X∂tρ,µ:=trt=0∂tEρ,µ,κ0>1−µ+1/p.
IntheNewtonpolygon,thesespacescanbeobtainedbyintersectingthehorizontallines
(a,1−µ+1/p)anda,1+(1−µ+1/p),a∈R,withitsnontrivialpart.Moreprecisely,
Theorem1.3.6isappliedtotheintersectionofthespacesthatdeterminetheedgesthese
horizontallinesintersect,respectively(cf.Figure1.3.2).
Thedescriptionofthespacesbelowfollowsthepresentationin[26].
ThenonredundantdescriptionofthespacesEρ,µ,Xρ,µandX∂tρ,µ.
Case1:l=2m.Onehas
Eρ,µ=Wp,µ1+κ0(J;Lp(Γ;F))∩Lp,µ(J;Wp2m(1+κ0)(Γ;F)),

118

MaximalLp,µ-RegularityforBoundaryConditionsofRelaxationType

thereforeandXρ,µ=B2p,pm(κ0+µ−1/p)(Γ;F),X∂tρ,µ=B2p,pm(κ0−(1−µ+1/p))(Γ;F)ifκ0>1−µ+1/p.
Case2:l<2m.Onehas
Eρ,µ=Wp,µ1+κ0(J;Lp(Γ;F))∩Lp,µ(J;Wpl+2mκ0(Γ;F))∩W1p,µ(J;Wp2mκ0(Γ;F)),
andthisyieldssimilartracespacesasinCase1,
Xρ,µ=B2p,pmκ0+l(µ−1/p)(Γ;F),X∂tρ,µ=B2p,pm(κ0−(1−µ+1/p))(Γ;F)ifκ0>1−µ+1/p.
Case3:l>2m.Thisisthemostcomplicatedcase.Onehas
Eρ,µ=Wp,µ1+κ0(J;Lp(Γ;F))∩Lp,µ(J;Wpl+2mκ0(Γ;F))∩Wκp,µj(J;Wpkj(Γ;F)),
∈JjwhereJ={j1,...,jqmax}⊂J,qmax∈N,containsthoseindicesj∈Jsothat(kj,κj)
belongstothenontrivialpartoftheNewtonpolygon,i.e.,thepoints
P0=(0,1+κ0),P1=(kj1,κj1),...,Pqmax=(kjqmax,κjqmax),
aretheverticesofitsnontrivialpart.Notethatitnecessarilyholdsljq>2mforjq∈J.
ItisassumedthatJisarrangedinawaysuchthat
kjq1<kjq2andκjq1>κjq2forq1<q2.
Forlaterconsiderationswedefine
k−1:=0,κ−1:=1+κ0,m−1:=m0−2m,l−1:=2m.
WefurtherdenotetheedgeintheNewtonpolygonconnectingthepointsPqandPq+1by
NPq,q=0,...,qmax,anddefine
J2q:=j∈J∪{−1}:(kj,κj)=Pq,q=0,...,qmax,
J2q+1:=j∈J∪{−1}:(kj,κj)∈NPq,q=0,...,qmax.
Thetemporaltracespaceof∂tρisobtainedbyTheorem1.3.6fromthespacescorresponding
toP0=(0,1+κ0)andP1=(kj1,κj1),i.e.,
X∂tρ,µ=Bkp,pj1(κ0−(1−µ+1/p))/(1+κ0−κj1)(Γ;F)ifκ0>1−µ+1/p.
NotethatTheorem1.3.6doesnotdirectlyapplyifκj1<1−µ+1/p.Inthiscaseonefirst
hastouseProposition1.3.2andthenapplyTheorem1.3.6tothespacescorrespondingto
thepoints(0,1+κ0)and(kj1κ0/(1+κ0−κj1),1),cf.Remark1.3.7.
ForXρ,µonehastodistinguishthreemorecases.
Case3(i):Ifκj>1−µ+1/pforallj∈J,then
Xρ,µ=Blp,p+2m(κ0−(1−µ+1/p))(Γ;F).

3.1TheProblemandtheApproachinWeightedSpaces

119

Case3(ii):Denotebyjq1∈Jbethesmallestindexwithκjq1>1−µ+1/p,andby
jq2∈Jthelargestindexwithκjq2<1−µ+1/p.Then
kjq1+(κjq1−(1−µ+1/p))κkjjqq2−−κkjjqq1
Xρ,µ=Bp,p21(Γ;F).
Case3(iii):Ifκj<1−µ+1/pforallj∈J,then
Xρ,µ=Bkp,pj1(κ0+µ−1/p)/(1+κ0−κj1)(Γ;F).

ItcanbeseenfromtheNewtonpolygonthatineachoftheCases1,2and3itholds
Xρ,µ→X∂tρ,µ.(3.1.5)
Wenowconsidercompatibilityconditionsattheboundaryatt=0,whicharenecessary
forthesolvabilityof(3.1.1).Forthedynamicequationontheboundary,ifκ0>1−µ+1/p
thenbyTheorem1.3.6itholds
F0,µ→CJ;B2p,pm(κ0−(1−µ+1/p))(Γ;F),
sothatthisequationhastoholduptot=0bycontinuity,providedthecoefficientsofB0
andC0aresufficientlysmooth.Inthiscaseitisthereforenecessarythat
g0(0,∙)−B0(0,∙,D)u0−C0(0,∙,DΓ)ρ0∈X∂tρ,µifκ0>1−µ+1/p,(3.1.6)
sinceotherwiseitisforallρ∈Eρ,µimpossibletosatisfythedynamicequationforthedata
g0,u0,ρ0.Moreover,ifκj>1−µ+1/pforsomej=1,...,m,thenitholdsasabove
Fj,µ→CJ;B2p,pm(κj−(1−µ+1/p))(Γ;E),
andalsothecorrespondingstaticboundaryequationsmustbevaliduptot=0byconti-
nuity.Hencethedatanecessarilysatisfies
Bj(0,∙,D)u0+Cj(0,∙,DΓ)ρ0=gj(0,∙)onΓifκj>1−µ+1/p,j=1,...,m,(3.1.7)
if(3.1.1)hasasolution(u,ρ)∈Eu,µ×Eρ,µ,providedthecoefficientsaresufficientlysmooth.

WeillustratethespacesEρ,µ,Xρ,µ,X∂tρ,µandthecompatibilityconditionsbyreconsidering
theproblemsfromExample3.1.1.
Example3.1.2.Problem(3.1.2)belongstoCase1,sincel=l0=2.Wehave
Eρ,µ=W3p,µ/2−1/2p(J;Lp(Ω;E))∩Lp,µ(J;Wp3−1/p(Ω;E)),
spacestracetheforandXρ,µ=B2(p,pµ−1/p)+1−1/p(Γ;F),X∂tρ,µ=B2(p,pµ−1/p)−1−1/p(Γ;F),

120

MaximalLp,µ-RegularityforBoundaryConditionsofRelaxationType

wherethetraceofthederivativeonlyexistsif2(µ−1/p)>1+1/p.Concerningthecompat-
ibilitycondition(3.1.6),notethatthetracespaceofF0,µequalsX∂tρ,µforκ0>1−µ+1/p.
Further,theoperators∂νand−ΔΓmapXu,µandXρ,µintoX∂tρ,µ,respectively.Hence
(3.1.6)isalwayssatisfied.Thecondition(3.1.7)requirestrΩu0=ρ0iftheseexpressions
exist,whichisnaturalfordynamicboundaryconditions.
Theproblem(3.1.3)belongstoCase2,sinceherel=l1=1<2.Therefore
Eρ,µ=W3p,µ/2−1/2p(J;Lp(Ω;E))∩Lp,µ(J;Wp2−1/p(Ω;E))∩W1p,µ(J;Wp1−1/p(Γ;F)),
furtherandXρ,µ=B(p,pµ−1/p)+1−1/p(Γ;F),X∂tρ,µ=B2(p,pµ−1/p)−1−1/p(Γ;F),
wherethelattertraceonlyexistsif2(µ−1/p)>1+1/p.Asabove(3.1.6)and(3.1.7)are
satisfied.naturallyFinally,theproblem(3.1.4)belongstoCase3,duetol=l1=3>2.Sincek0=−∞it
holdsEρ,µ=W3p,µ/2−1/2p(J;Lp(Γ;F))∩Lp,µ(J;Wp4−1/p(Γ;F))∩W1p,µ−1/p(J;Wp2(Γ;F)).
Thetracespaceof∂tρ,whichexistsfor2(µ−1/p)>1+1/p,isgivenby
X∂tρ,µ=B4(p,pµ−1/p)−2−2/p(Γ;F).
Forthetracespaceofρ,ifµ>3/2pthenweareinCase3(i)andobtain
Xρ,µ=B2(p,pµ−1/p)+2−1/p(Γ;F),
andifµ<3/2pthenweareinCase3(iii)with
Xρ,µ=B4(p,pµ−1/p)+2−2/p(Γ;F).
Thisshowsthattheinitialregularityforρcanchangedrasticallyifµvaries.TheCase
3(ii)cannotoccurinthisexample,sincethereisonlyonenontrivialvertexintheNewton
polygon.Forµ>1/2+3/2p,i.e.,2(µ−1/p)>1+1/p,thecondition(3.1.6)isnotalways
satisfied,sinceforg0∈F0,µitholds,ingeneral,g0(0,∙)∈B2(p,pµ−1/p)−1−1/p(Γ;F),andthe
latterspacehasalwaysalowerregularitythanX∂tρ,µ.Finally,for(3.1.7)thedatamust
satisfyu0(∙)+ΔΓρ0(∙)=g1(0,∙)ifµ>3/2p.
Weintendtosolve(3.1.1)inthefollowingsense.
Definition3.1.3.Wesaythat(3.1.1)enjoysthepropertyofmaximalLp,µ-regularityon
theintervalJ=(0,T)iftheregularityassumptionsonthedata,i.e.,
f∈E0,µ,g∈Fµ,u0∈Xu,µ,ρ0∈Xρ,µ,
togetherwiththecompatibilityconditions(3.1.6)and(3.1.7),arenotonlynecessaryfora
uniquesolution(u,ρ)∈Eu,µ×Eρ,µof(3.1.1),butalsosufficient.

3.1TheProblemandtheApproachinWeightedSpaces

121

TheAssumptionsontheOperators
Inthesequel,thesubscriptdenotestheprinciplepartofadifferentialoperator,withan
importantexceptionfortheCj,forwhichweset
Cj:=0ifj∈/J.
HenceonlytheprinciplepartsoftheoperatorsCjcorrespondingtoapointonthenontrivial
partoftheNewtonpolygonforEρ,µareconsidered.
First,thecoefficientsoftheoperatorsarerequiredtobesuchthateachsummandoccurring
inA,BandCisacontinuousoperatorontherespectiveunderlyingspaces.Moreover,for
localizationpurposesitisrequiredthatthetopordercoefficientsoftheoperatorsare
boundedanduniformlycontinuous.Asinthestaticcase,thePropositions1.3.15and
1.3.24showthatforthecoefficientsofAandBthefollowingissufficientforourpurposes,
.elyectivresp(SD)For|α|<2moneofthefollowingtwoconditionsisvalid:either
2m(µ−1/p)>2m−1+n/pandaα∈E0,µ(J×Ω;B(E)),
ortherearerα,sα∈[p,∞)withp(1−sαµ)+1+2mrnα<1−2|αm|suchthat
aα∈LsαJ;(Lrα+L∞)(Ω;B(E)).
For|α|=2mitholdsaα∈BUC(J×Ω;B(E)),andifΩisunboundedtheninaddition
thelimitsaα(t,∞):=lim|x|→∞aα(t,x)existuniformlyint∈J.
(SB)LetE0=B(E,F),andEj=B(E)forj=1,...,m.Forj=0,...,mand|β|≤mjone
ofthefollowingtwoconditionsisvalid:either
κj>1−µ+1/p+n−1andbjβ∈Fj,µ,
mp2ortherearerjbβ,sbjβ∈[p,∞)with
p(1−bµ)+1+n−b1<κj+mj−|β|,κj+mj−|β|−(1−µ+1/p)∈/0,n−b1,
sjβ2mrjβ2m2m2mrjβ
suchthat
bjβ∈Bsκjbjβ,pJ;Lrjbβ(Γ;Ej)∩LsjbβJ;Br2jbβmκ,pj(Γ;Ej).
ForthelocalcoefficientsofC,thefollowingconditionsaresufficient,asLemma3.2.4shows.1
1correspAsonthedingproofonesofinLemma(SB).Dep3.2.4endingshows,onthetheregularitrelationybetweenassumptionstheinspaces(SC)Eρ,µareandnotFµ,asthesharpcoasefficienthets
couldbelessregularintheCases2and3.Forthesakeofaunifiedpresentationwedonotdistinguish
betweenthethreecases.

122

MaximalLp,µ-RegularityforBoundaryConditionsofRelaxationType

(SC)LetF0=B(F)andFj=B(F,E)forj=1,...,m,andletg:V→Γbeany
coordinatesforΓ.Forj=0,...,mand|γ|≤kjoneoffollowingtwoconditionsis
eitheralid:vκj>1−µ+1/p+n−1andcjgγ∈Fj,µ(J×V;Fj),
mp2ortherearerjcγ,sjcγ∈[p,∞)thatsatisfy
p(1−µ)+1n−1kj−|γ|
sjcγ<1−(2mκj+kj−|γ|)rjcγκj+kj(1+κ0−κj)
and,incaseκj>1−µ+1/p,
1−µ+1/pn−1
1−κj+kj−|γ|(1+κ0−κj)>(2mκj+kj−|γ|)rjcγ,
kjthathsuccjgγ∈Bsκjcjγ,p(J;Lrjcγ(V;Fj))∩Lsjcγ(J;Br2jcγmκ,pj(V;Fj)).
Notethatthesecondconditiononthetopordercoefficientsisequivalenttop(1s−cµ)+1+
γj2nmr−jc1γ<κj,whichisthesameasin(SB).Proposition1.3.24andLemma3.2.4showthat
(SB)and(SC)implythecontinuityofthe(local)topordercoefficients,i.e.,
bjβ∈BUC(J×Γ;Ej),|β|=mj,cjgγ∈BUC(J×V;Fj),|γ|=kj,j=0,...,m.
Wenextdescribethestructuralassumptionson(A,B,C).Asinthestaticcase,forAwe
assumenormalellipticity.
(E)Forallt∈J,x∈Ωand|ξ|=1itholdsσ(A(t,x,ξ))⊂C+.IfΩisunboundedthen
itholdsinadditionσ(A(t,∞,ξ))⊂C+forallt∈Jand|ξ|=1.
AlsoconditionsofLopatinskii-Shapirotypearerequired.Wecalllocalcoordinatesgasso-
ciatedtox∈Γifthecorrespondingchart(U,ϕ)satisfies
ϕ(x)=0,ϕ(x)=Oν(x),ϕ(U∩Ω)⊂R+n,ϕ(U∩Γ)⊂Rn−1×{0},
whereOν(x)isafixedorthogonalmatrixthatrotatesν(x)to(0,...,0,−1)∈Rn.Sucha
chart(U,ϕ)existsbyLemmaA.1.1.Forsuchcoordinateswedefinetherotatedoperators
AνandBνby
Aν(t,x,D):=A(t,x,OνT(x)D),Bν(t,x,D):=B(t,x,OνT(x)D).
Moreover,wedefinethelocaloperatorCgwithrespecttogby
Cjg(t,x,Dn−1):=cjgγ(t,g−1(x))Dnγ−1,j=0,...,m,
k|≤γ|jwherecjgγarethelocalcoefficientsofCj.Withthesenotations,ineachoftheCases1,2
and3weassumethefollowing.2RecallthatCjg:=0forj∈/J.
2ThereadershouldnotbeterrifiedoftheLopatinskii-Shapiroconditions.Itisnottoohardtoverify
theminapplications,cf.[26,Section3]andSection5.2.

3.1TheProblemandtheApproachinWeightedSpaces

123

(LS)Foreachfixedx∈Γ,choosecoordinatesgassociatedtox.Thenforallt∈J,all
λ∈C+andξ∈Rn−1with|λ|+|ξ|=0,allh0∈Fandallhj∈E,j=1,...,m,the
ordinaryinitialvalueproblem
λ+Aν(t,x,ξ,Dy)v(y)=0,y>0,
B0ν(t,x,ξ,Dy)v|y=0+λ+C0g(t,x,ξ)σ=h0,
Bjν(t,x,ξ,Dy)v|y=0+Cjg(t,x,ξ)σ=hj,j=1,...,m,
hasauniquesolution(v,σ)∈C0([0,∞);E)×F.
ForproblemsfromCase2and3,inadditionthefollowingasymptoticLopatinskii-Shapiro
conditions(LS∞−)and(LS∞+)arerequired,respectively.
(LS−∞)Letl<2m.Foreachfixedx∈Γ,choosecoordinatesgassociatedtox.Thenfor
allt∈J,allh0∈F,allhj∈E,j=1,...,m,andallλ∈C+,ξ∈Rn−1with
|λ|+|ξ|=0,theordinaryinitialvalueproblem
λ+Aν(t,x,ξ,Dy)v(y)=0,y>0,
Bjν(t,x,ξ,Dy)v|y=0=hj,j=1,...,m,
andforallλ∈C+and|ξ|=1theproblem
Aν(t,x,ξ,Dy)v(y)=0,y>0,
B0ν(t,x,ξ,Dy)v|y=0+λ+C0g(t,x,ξ)σ=h0,
Bjν(t,x,ξ,Dy)v|y=0+Cjg(t,x,ξ)σ=hj,j=1,...,m,
hasauniquesolution(v,σ)∈C0([0,∞);E)×F,respectively.
(LS+∞)Letl>2m.Foreachfixedx∈Γ,choosecoordinatesgassociatedtox.Thenforall
t∈J,allh0∈F,allhj∈E,j=1,...,m,andallλ∈C+andξ∈Rn−1\{0},the
ordinaryinitialvalueproblem
λ+Aν(t,x,ξ,Dy)v(y)=0,y>0,
Bjν(t,x,ξ,Dy)v|y=0+δj,J2qmax+1Cjg(t,x,ξ)σ=hj,j=0,...,m,
andfurtherforallλ∈C+\{0},|ξ|=1andq=1,...,2qmax,theproblem
λ+Aν(t,x,0,Dy)v(y)=0,y>0,
B0ν(t,x,0,Dy)v|y=0+δ−1,Jqλσ+δ0,JqC0g(t,x,ξ)σ=h0,
Bjν(t,x,0,Dy)v|y=0+δj,JqCjg(t,x,ξ)σ=hj,j=1,...,m,
hasauniquesolution(v,σ)∈C0([0,∞);E)×F,respectively.Hereδj,Jq=1ifj∈Jq
andδj,Jq=0otherwise.
IfEandFarefinitedimensional,itsufficestoshowthattheaboveproblemswithhj=0,
j=0,...,m,admitonlythetrivialsolutions,respectively.In[26]itisshownthatthese
conditionsarenecessaryforuniformmaximalLp-regularityof(3.1.1)onfiniteintervals.The
Lopatinskii-Shapiroconditionsareverifiedin[26,Section3]fortheproblemsofExample
3.1.1.WealsorefertoSection5.2,whereweverify(LS)foramoregeneralversionof
(3.1.2).

124

MaximalLp,µ-RegularityforBoundaryConditionsofRelaxationType

TheoremMainTheWestatethemainresultofthischapter.
Theorem3.1.4.LetEandFbeBanachspacesofclassHT,p∈(1,∞)andµ∈(1/p,1].
LetJ=(0,T)beafiniteinterval,T>0,andletΩ⊂Rnbeadomainwithcompact
smoothboundaryΓ=∂Ω.Assumethat(A,B,C)satisfies(E),(LS),(SD),(SB)and(SC),
and,inaddition,forl<2mcondition(LS∞−),andforl>2mcondition(LS∞+).Assume
furtherthatκj=1−µ+1/pforj=0,...,m.Thentheproblem
∂tu+A(t,x,D)u=f(t,x),x∈Ω,t∈J,
∂tρ+B0(t,x,D)u+C0(t,x,DΓ)ρ=g0(t,x),x∈Γ,t∈J,
Bj(t,x,D)u+Cj(t,x,DΓ)ρ=gj(t,x),x∈Γ,t∈J,j=1,...,m,(3.1.8)
u(0,x)=u0(x),x∈Ω,
ρ(0,x)=ρ0(x),x∈Γ,
enjoysmaximalLp,µ-regularity,i.e.,(3.1.8)hasauniquesolution(u,ρ)∈Eu,µ×Eρ,µif
andonlyif(f,g,u0,ρ0)∈D,where
D:=(f,g,u0,ρ0)∈E0,µ×Fµ×Xu,µ×Xρ,µ:forj=1,...,mitholds
Bj(0,∙,D)u0+Cj(0,∙,DΓ)ρ0=gj(0,∙)onΓifκj>1−µ+1/p;
g0(0,∙)−B0(0,∙,D)u0−C0(0,∙,DΓ)ρ0∈X∂tρ,µifκ0>1−µ+1/p.
ThecorrespondingsolutionoperatorL:D→Eu,µ×Eρ,µiscontinuous.RestrictingLto
D0:=(f,g,u0,ρ0)∈D:g∈0Fµ,
forgivenT0>0itsoperatornormisuniforminT≤T0.Finally,ifthecoefficients
(−i)|α|aα,|α|≤2m,(−i)|β|bjβ,|β|≤mj,(−i)|γ|cjgγ,|γ|≤kj,j=0,...,m,
andthedataarereal-valued,thenalsothesolutionisreal-valued.
HerethesetsofcompatibledataDandD0areendowedwiththenorms
|(f,g,u0,ρ0)|D:=|f|E0,µ+|g|Fµ+|u0|Xu,µ+|ρ0|Xρ,µ
+|g0(0,∙)−B0(0,∙,D)u0−C0(0,∙,DΓ)ρ0|X∂tρ,µ,
and|(f,g,u0,ρ0)|D0:=|f|E0,µ+|g|0Fµ+|u0|Xu,µ+|ρ0|Xρ,µ
+|g0(0,∙)−B0(0,∙,D)u0−C0(0,∙,DΓ)ρ0|X∂tρ,µ,
respectively.ThecontinuityofLmustbeunderstoodwithrespecttothesenorms.Againit
isimportanttodistinguishbetweenthenormsofFµand0Fµ(seeRemark1.1.15).Arguing
asintheproofofLemma1.3.25,onecanshowthat(D,|∙|D)and(D0,|∙|D0)areBanach
spaces.

3.2Half-SpaceProblemswithBoundaryConditions

125

WehaveseeninExample3.1.2thatitispossiblethatthecondition
g0(0,∙)−B0(0,∙,D)u0−C0(0,∙,DΓ)ρ0∈X∂tρ,µ
isalwayssatisfied.InthiscasewemayconsiderDandD0asclosedsubspacesofE0,µ×
Fµ×Xu,µ×Xρ,µandE0,µ×0Fµ×Xu,µ×Xρ,µ,respectively.
TheproofofTheorem3.1.4follows[26]andisbasedonalocalizationandperturbation
procedure,analogouslytotheproofofTheorem2.1.4.Forageneraloutlinewerefertothe
endofSection2.1.Themaindifficultyisthehalf-spacecaseonthehalf-line,withvanishing
initialvalueandvanishingdomaininhomogeneity.Sincetherearenoboundaryconditions
involvedinthefullspaceproblem,thiscaseisalreadycoveredbythePropositions2.2.2
2.3.2.and

3.2Half-SpaceProblemswithBoundaryConditions
3.2.1ConstantCoefficients
Onthehalf-spaceΩ=R+nwithboundaryΓ=Rn−1weconsiderthehomogeneousdiffer-
eratoroptialenA(D)=aαDα,
m=2|α|togetherwiththehomogeneousboundaryoperators
Bj(D)=bjβtrR+nDβ,Cj(Dn−1)=cjγDnγ−1j=0,...,m.
|β|=mj|γ|=kj
Thecoefficientsoftheoperatorsareassumedtobeconstant,respectively,
aα,bjβ∈B(E),cjγ∈B(F,E),j=1,...,m,b0β∈B(E,F),c0γ∈B(F).
Inthissection,ifnothingelseisindicated,allspaceshavetobeunderstoodoverR+×R+n,
oroverR+×Rn−1.Weset
0Fj,µ:=0Wκp,µj(R+;Lp(Rn−1;E))∩Lp,µ(R+;Wp2mκj(Rn−1;E)),j=1,...,m,
andanalogouslyforthespaces0F0,µ,0Fµ,0Eu,µ,and0Eρ,µ.3

Wefirstconsiderinhomogeneousboundaryconditions.Theprooffollows[26,Section4.3].
Lemma3.2.1.LetEandFbeBanachspacesofclassHT,p∈(1,∞),µ∈(1/p,1],
andassumethat(A,B,C)satisfies(E)and(LS),and,inaddition,forl<2mcondition
3Moreprecisely,eachWsp,µ-spaceinthenonredundantdescriptionof0Eρ,µmustbeequippedwitha
subleft.0script

126

MaximalLp,µ-RegularityforBoundaryConditionsofRelaxationType

(LS∞−),andforl>2mcondition(LS∞+).Thenforg∈0Fµthereisauniquesolution
(u,ρ)∈0Eu,µ×0Eρ,µof
u+∂tu+A(D)u=0,x∈R+n,t>0,
ρ+∂tρ+B0(D)u+C0(Dn−1)ρ=g0(t,x),x∈Rn−1,t>0,
Bj(D)u+Cj(Dn−1)ρ=gj(t,x),x∈Rn−1,t>0,j=1,...,m,(3.2.1)
u(0,x)=0,x∈R+n,
ρ(0,x)=0,x∈Rn−1.
Proof.(I)Wefirstshowuniquenessfor(3.2.1).OnthespaceLp(R+n;E)×Wps(Rn−1;F),
κkwheres=2mκ0intheCases1and2,ands=1+κj01−0κj1inCase3,4weconsiderthe
operatorA,definedby
A(u,ρ):=(1+A)u,B0u+(1+C0)ρ,(u,ρ)∈D(A),
domainwithD(A):=(u,ρ)∈Wp2m(R+n;E)×Wpl+2mκ0(Rn−1;F):
Bju+(1+Cj)ρ=0,j=1,...,m;B0u+C0ρ∈Wps(Rn−1;F).
By(theproofof)[26,Theorem2.2],AgeneratesananalyticC0-semigroup,anditthus
followsfromLemma1.2.1thatsolutionsof(3.2.1)areuniqueinthemaximalLp,µ-regularity
spaceforA,i.e.,in
Eu,µ(R+)×W1p,µ(R+;Wps(Rn−1;F))∩Lp,µ(R+;Wpl+2mκ0(Rn−1;F)).
Since0Eu,µ×0Eρ,µembedsintothisspace,itfollowsthatsolutionsof(3.2.1)areunique
in0Eu,µ×0Eρ,µ.
(II)Therestofthisproofisconcernedwiththeexistenceofsolutionsof(3.2.1).Wefirst
thatosesuppg=(g0,...,gm)∈Cc∞R+;Wp2m(Rn−1;F×Em).
WeapplytheFouriertransformFtintimeto(3.2.1),withcovariableθ∈R,toarriveat
lemprobstationarythe(1+iθ)v+A(D)v=0,x∈R+n,
(1+iθ)σ+B0(D)v+C0(Dn−1)σ=Ftg0(θ),x∈Rn−1,(3.2.2)
Bj(D)v+Cj(Dn−1)σ=Ftgj(θ),x∈Rn−1,j=1,...,m.
In[26,Section4.3]itisshownthat(3.2.2)hasforeachθ∈Rauniquesolution
v(θ),σ(θ)∈Wp2m(R+n;E)×Wpl+2mκ0(Rn−1;F),
4ThenumbersisdeterminedbytheintersectionofthenontrivialpartoftheNewtonpolygonofEρ,µ
withtheverticalline(a,1),a∈R.Thenumberj1∈JwasdefinedinSection3.1.

3.2Half-SpaceProblemswithBoundaryConditions

127

whichmayberepresentedasfollows.Wewritex=(x,y)∈R+nwithx∈Rn−1and
y>0,anddenotebyFxthepartialFouriertransformwithrespecttox,withcovariable
ξ∈Rn−1.Wefurtherdefinethesymbols
ϑ:=(1+iθ+|ξ|2m)1/2m,b:=|ξ|,ζ:=ξ,a:=12+miθ,
ϑξϑ||andtheso-calledboundarysymbols(θ,ξ)by
s(θ,ξ):=1+iθ+|ξ|lintheCases1and2,
s(θ,ξ):=1+iθ+|ξ|kjϑm0−mjinCase3.
∈JjSettingh(θ,ξ):=ϑ−m0(FxFtg0)(θ,ξ),...,ϑ−mm(FxFtgm)(θ,ξ)∈F×Em,itholds
v(θ,x,y)=firstcomponentofFx−1eϑiA0(bζ,a)yPs(bζ,a)Mu0(b,ζ,ϑ)h(θ,∙)
σ(θ,x,y)=Fx−1s(∙,θ)−1ϑm0Mρ0(b,ζ,ϑ)h(θ,∙).
Herewehave
A0:Cn−1×C→B(E2m),Ps:Cn−1×C→B(E2m),
Mu0:Db×Dζ×Σ→B(F×Em,E2m),Mρ0:Db×Dζ×Σ→B(F×Em,F),
setsenopwith(B1/2(1/2))1/2m⊂Db⊂C,{ζ∈Rn−1:|ζ|=1}⊂Dζ⊂Cn−1,
andasectorΣ=z∈C\{0}:|argz|<φ,whereφ∈(π/4m,π).ThemapsA0,Ps,
Mu0andMρ0areholomorphicintheircomplexarguments.ThespectrumofiA0(bζ,a)has
agapattheimaginaryaxis,andPs(bζ,a)isthespectralprojectioncorrespondingtothe
stablepartofthespectrum.ThefunctionsMu0andMρ0havethepropertythat
|ξ||α|DξαMu0(b,ζ,ϑ):α∈{0,1}n−1,ξ=0,θ∈R,b∈Db,ϑ∈Σ(3.2.3)
isanR-boundedsetofoperatorsinB(F×Em,E2m),andthat
|ξ||α|DξαMρ0(b,ζ,ϑ):α∈{0,1}n−1,ξ=0,θ∈R,b∈Db,ϑ∈Σ(3.2.4)
isanR-boundedsetofoperatorsinB(F×Em,F).
Therepresentationofthesolutionisobtainedin[26]byapplyingFxto(3.2.2),which
yieldsanordinaryinitialvalueproblem.By(LS),thisproblemhasauniquedecaying
solution,fromwhichthesolutionof(3.2.2)isobtainedbyapplyingFx−1.Theasymptotic
Lopatinskii-Shapiroconditions(LS∞−)and(LS∞+)arerequiredtoshowtheR-boundedness
ofthesesets,duetotheunboundednessofϑ.Forl=2m,thesymbolsMu0andMρ0donot
dependonϑ,sothatinthiscaseasymptoticconditionsarenotneeded.Wereferto[26,
details.for4.3]Section

128

MaximalLp,µ-RegularityforBoundaryConditionsofRelaxationType

(III)Wederiveanotherrepresentationofthesolutionoperatorfor(3.2.2).Forafunction
h∈S(Rn−1;Em)wecalculateforx∈Rn−1andy>0,neglectingtheargumentsofA0
,PandsFx−1eiϑA0yPsh(x)=Fx−1eiϑA0(y+ye)Pse−yeϑh(x)|ye=0(3.2.5)
∞=−∂yeFx−1eiϑA0(y+ye)Pse−yeϑh(x)dy
0=∞Fx−1eiϑA0(y+ye)Ps1−iA0ϑ2me−yeϑh(x)dy
∞0ϑ2m−1
ϑ=Fx−1eiϑA0(y+ye)Ps12−mi−A10∗(LθEθFx−1h)(∙,y)(x)dy.
0HeretheoperatorLθ,whichcorrespondstothesymbolϑ2m,isdefinedby
Lθ:=1+iθ+(−Δn−1)m,
andtheextensionoperatorEθ,whichcorrespondstoy→e−yϑ,isforf∈Lp(Rn−1;E)
ybengiv(Eθf)(x,y):=e−yLθ1/2mf(x),x∈Rn−1,y>0.
Inthelastlineof(3.2.5)wehaveusedthat
Fx−1ϑ2me−∙ϑh=LθEθFx−1h,h∈S(Rn−1;Em),
whichisaconsequenceoftheuniquenessoftheH∞-calculusfor−Δn−1onLp(Rn−1;E)
(see[62,Example10.2]).Forθ∈Randf∈Lp(R+n;E2m)wethusdefinetheoperatorT(θ)
yb∞(T(θ)f)(x,y):=firstcomponentofFx−1eiϑA0(y+ye)Ps1ϑ2−mi−A10∗f(∙,y(x)dy.
0TheproofsoftheLemmas4.3and4.4in[25]show
T∈C1R;B(Lp(R+n;E2m),Wp2m(R+n;E)),
thatandDαT(θ),θ∂∂θDαT(θ):θ∈R,|α|≤2m(3.2.6)
isanR-boundedsetofoperatorsinBLp(R+n;E2m),Lp(R+n;E).Usingfurtherthat
ϑ−mjFx=FxLθ−mj/2mforj=0,...,m,whichcanbeseenasabove,forthecomponent
v(θ)ofthesolutionof(3.2.2)wethereforearriveattherepresentation
v(θ)=T(θ)LθEθFx−1Mu(b,ζ,ϑ)FxLθ−mj/2mFtgj(θ)j=0,...,m.
Similarly,thesecondcomponentσ(θ)mayberepresentedby
σ(θ)=Sθ−1Lθm0/2mFx−1Mρ0(b,ζ,ϑ)FxLθ−mj/2mFtgj(θ)j=0,...,m,
wheretheoperatorSθ,thatcorrespondstotheboundarysymbols(θ,ξ),isgivenby
Sθ:=1+iθ+(−Δn−1)l/2intheCases1and2,

3.2Half-SpaceProblemswithBoundaryConditions129
Sθ:=1+iθ+(−Δn−1)kj/2Lθ(m0−mj)/2minCase3.
∈Jj(IV)Itfollowsfromtheboundednessof(3.2.6)thatthemapθ→T(θ)isbounded.Itwas
furthershowninStepIIoftheproofofLemma2.2.7thatθ→LθEθisbounded.Moreover,
theR-boundednessof(3.2.3)and(3.2.4),theoperator-valuedFouriermultipliertheorem
inn−1dimensions([24,Theorem3.25],seealso[62,Theorem4.13])andrealinterpolation
mapsthealsothatyieldθ→Fx−1Mu0(b,ζ,ϑ)Fx∈BWps(Rn−1;F×Em),Wps(Rn−1;E2m),
θ→Fx−1Mρ0(b,ζ,ϑ)Fx∈BWps(Rn−1;F×Em),Wps(Rn−1;F),
areboundedforθ∈R.Henceforg∈Cc∞R+;Wp2m(Rn−1;F×Em)wemayapplythe
inverseFouriertransformtov(θ),σ(θ)andobtainthatthesolution(u,ρ)of(3.2.1)is
ybengivu=Lug:=Ft−1T(θ)LθEθFx−1Mu0(b,ζ,ϑ)FxLθ−mj/2mFtgjj=0,...,m,
ρ=Lρg:=Ft−1Sθ−1Lθm0/2mFx−1Mρ0(b,ζ,ϑ)FxLθ−mj/2mFtgj.
,...,m=0jNowasinStepIIIoftheproofofLemma2.2.7wemayrewritethesesolutionoperatorsto

Lug=Ft−1T(θ)FtLEFt−1Fx−1M0u(b,ζ,ϑ)FxFtL−mj/2mgjj=0,...,m,
Lρg=S−1Lm0/2mFt−1Fx−1Mρ0(b,ζ,ϑ)FxFtL−mj/2mgjj=0,...,m,
eratorsopthewithL:=1+∂t+(−Δn−1)m,E:=e−∙L1/2m,
S:=1+∂t+(−Δn−1)l/2intheCases1and2,
S:=1+∂t+(−Δn−1)kj/2L(m0−mj)/2minCase3,
∈JjthatcorrespondtoLθ,EθandSθ,respectively.SinceCc∞(R+;Wp2m(Rn−1;F×Em))isa
densesubsetof0FµbyLemma1.3.14,thetaskisnowtoshowthatthereisanestimate
|Lug|Eu,µ+|Lρg|Eρ,µ|g|0Fµ,g∈Cc∞(R+;Wp2m(Rn−1;F×Em)),(3.2.7)
sincethenthesolutionoperator
L:=(Lu,Lρ)
extendscontinuouslyto0Fµ,andthisyieldsthesolutionof(3.2.1).
(V)ByLemma1.3.1,therealizationofLonthespaceLp,µ(R+;Lp(Rn−1;E))isinvertible,
sectorialofanglenotlargerthanπ/2,andfors∈(0,1]wehave
DL(s,p)=0Wsp,µ(R+;Lp(Rn−1;E))∩Lp,µ(R+;Wp2ms(Rn−1;E)).

130

MaximalLp,µ-RegularityforBoundaryConditionsofRelaxationType

ThereforeL−mj/2mmapsforj=1,...,mthespace0Fj,µ=DL(κj,p)continuouslyinto
0YE:=DL(1−1/2mp,p)=0W1p,µ−1/2mp(R+;Lp(Rn−1;E))
∩Lp,µ(R+;Wp2m−1/p(Rn−1;E)).
ThesameargumentsshowthatL−m0/2mmaps0F0,µcontinuouslyinto0YF,whichis
definedas0YEwithEreplacedbyF.
(VI)Wenextshowthattheoperator
M0:=Ft−1Fx−1M0(b,ζ,ϑ)FxFt,
withsymbolM0:Db×Dζ×Σ→B(F×Em,E2m×F)givenby
M0(b,ζ,ϑ):=Mu0(b,ζ,ϑ),Mρ0(b,ζ,ϑ),
extendscontinuouslytoanelementofB0YF×0YEm,0YE2m×0YF.Tothisendweconsider
theapproximatingoperators
M0,ε:=Ft−1Fx−1M0(b,ζ,ϑ)(1+ϑ)−εFxFt,ε∈(0,1).
Cauchy’sformulayieldstherepresentation
4πΞϑΞb
M0,ε=−12Ft−1Fx−1M0(b,ζ,ϑ)(1+ϑ)−ε(b−b)−1(ϑ−ϑ)−1FxFtdbdϑ,
withΞϑ=(−∞,0]e−iφ∗∪[0,∞)eiφ∗forsomeφ∗∈(π/4m,φ),andwhereΞbisaclosed
curveinDbsurrounding(B1/2(1/2))1/2m.Sinceζ=ξ/|ξ|isindependentofθ,wemay
tothisrewriteΞΞM0,ε=−4π12Fx−1M0(b,ζ,ϑ)Fx(1+ϑ)−ε(b−B)−1(ϑ−L1/2m)−1dbdϑ,
bϑwhereB:=(−Δn−1)1/2L−1/2mcorrespondstothesymbolb=|ξϑ|.Therealizationof
BonLp,µ(R+;Lp(Rn−1;E))isaboundedoperator,anditsspectrumiscontainedin
(B1/2(1/2))1/2m.Thiscanbeseenusingthejointfunctionalcalculusfor∂tand(−Δn−1)m
4.5].Theorem[57,AsaboveitfollowsfromtheR-boundednessofthesets(3.2.3)and(3.2.4),andtheoperator-
valuedFourier-multipliertheoreminRn−1thattheoperators
M1(b,ϑ):=Fx−1M0(b,∙,ϑ)Fx,b∈Db,ϑ∈Σ,
extendcontinuouslytoelementsofBWps(Rn−1;F×Em),Wps(Rn−1;E2m×F),s≥0,with
uniformlyboundedoperatorsnorms.SinceM0isholomorphic,alsoM1isholomorphicin
itsarguments.Bycanonicalpointwiseextensionwethusobtainthat
M1:Db×Σ→B0YF×0YEm,0YE2m×0YF
isboundedandholomorphic.HencewemayrewriteM0,εinto
M0,ε=−4π12M1(b,ϑ)(1+ϑ)−ε(b−B)−1(ϑ−L1/2m)−1dbdϑ,
ΞΞbϑ

3.2Half-SpaceProblemswithBoundaryConditions131
wherethecurveintegralsarenowinB0YF×0YEm,0YE2m×0YF.UsingL1−1/2mpas
anisomorphism0YE→Lp,µ(R+;Lp(Rn−1;E))thatcommuteswithB,weseethatthe
spectrumoftherealizationofBon0YF×0YEmisalsocontainedin(B1/2(1/2))1/2m.Now
theDunfordcalculusforByields
iπ2M0,ε=1M2(ϑ)(1+ϑ)−ε(ϑ−L1/2m)−1dϑ,
Ξϑwithaboundedholomorphicmap
M2:Σ→B0YF×0YEm,0YE2m×0YF.
SincetherealizationofL1/2monLp,µ(R+;Lp(Rn−1;E))issectorialwithanglenotlarger
thanπ/4m,itfollowsfrom[18,Corollary1]thatL1/2madmitsanoperator-valuedbounded
H∞-calculuswithH∞-anglenotlargerthanπ/4montherealinterpolationspaces0YEm
and0YFm.FromthisfactandtheboundednessofM2onΣweinfer
|M0,ε|B(0YF×0YEm,0YE2m×0YF)esup|M2(ϑ)(1+ϑ)−ε|B(0YF×0YEm,0YE2m×0YF)≤C,(3.2.8)
Σ∈ϑwhereCdoesnotdependonε∈(0,1).Dueto[24,Proposition2.2],forh∈D(L2)themap
ε→(1+L1/2m)εhiscontinuouswithvaluesinDL(1−1/2mp,p).Togetherwith(3.2.8),
yieldsfactthis|M0h|0YE2m×0YFlimsup|M0,ε|B(0YF×0YEm,0YE2m×0YF)|(1+L1/2m)εh|0YF×0YEm
0→ε|h|0YF×0YEm.
SinceD(L2)isdenseinDL(1−1/2mp,p),weobtainthatM0extendstoanelementof
B0YF×0YEm,0YE2m×0YF,asasserted.
(VII)NowwecanshowtherequiredestimateforLu,i.e.,
|Lug|Eu,µ|g|0Fµ,g∈Cc∞(R+;Wp2m(Rn−1;F×Em)).(3.2.9)
ByLemma1.3.8,theextensionoperatorE=e−∙L1/2mmapscontinuously
DL(1−1/2mp,p)=DL1/2m(2m−1/p,p)→Lp(R+;D(L)),
andLmapsthespaceLp(R+;D(L))continuouslyinto
LpR+;Lp,µ(R+;Lp(Rn−1;E))=Lp,µ(R+;Lp(R+n;E)).
Ofcourse,hereEmaybereplacedbyF.ThusLEmapscontinuously
0YE2m×0YF→Lp,µ(R+;Lp(R+n;E2m×F)).
TheR-boundednessof(3.2.6)andtheoperator-valuedFouriermultipliertheoreminLp,µ
(Theorem1.2.4)showthatFt−1TFtextendstoacontinuousoperator
Lp,µ(R+;Lp(R+n;E2m))→Lp,µ(R+;Wp2m(R+n;E)).5
5Asinthestaticcase,followingthemethodsoftheproofof[25,Lemma4.4],onecanshowthatfor
|α|≤2mitholdsDααTe∈C2(R;B(Lp(R+n;E2m×F)),andthat∂θ2|DαTj(θ)|θ12.HencealsoProposition
1.2.5yieldsthatDTeisaFouriermultiplieronLp,µ.

132

MaximalLp,µ-RegularityforBoundaryConditionsofRelaxationType

NowtheequationforushowsthattheEu,µnormofucanbecontrolledbyits
Lp,µ(R+;Wp2m(R+n;E))-norm,andthisyields(3.2.9).
(VIII)WefinallyconsidertheestimateforLρ,
|Lρg|Eu,µ|g|0Fµ,g∈Cc∞(R+;Wp2m(Rn−1;F×Em)).
AsaboveweobtainthattheoperatorLm0/2mmapscontinuously
0YF=DL(1−1/p,p)→DL(κ0,p)=0F0,µ.
ItisthuslefttoshowthatSisanisomorphism0Eρ,µ→0F0,µ.Usingthat∂tadmits
aboundedH∞-calculusonLp,µ(R+;Lp(Rn−1;F))withdomain0W1p,µ(R+;Lp(Rn−1;F))
(Theorem1.1.7),thiscanbedoneliterallyinthesamewayasin[26,Section4.2].
Beforeweturntothegeneralhalf-spacecaseweconsideraright-inverseforthetemporal
.Eontraceρ,µLemma3.2.2.LetFbeaBanachspaceofclassHT,p∈(1,∞),µ∈(1/p,1],andlet
Ω⊂RnbeadomainwithcompactsmoothboundaryΓ=∂Ω,orΩ=R+n.Assumethat
κ0=1−µ+1/p,andletρ0∈Xρ,µ(Γ)andfurtherρ1∈X∂tρ,µ(Γ)incaseκ0>1−µ+1/p
begiven.Thenthereisρ∈Eρ,µ(R+×Γ)with
ρ|t=0=ρ0,and∂tρ|t=0=ρ1ifκ0>1−µ+1/p.
Proof.(I)Weset
ρ1:=0forκ0<1−µ+1/p.
FirstsupposethatΓ=Rn−1.LetB0andB1benegativegeneratorsofexponentiallystable
analyticC0-semigroupsonLp(Rn−1;F).Thenwedefineρby
ρ(t)=2e−tB0−e−2tB0ρ0+e−tB1−e−2tB1B1−1ρ1,t>0,
sothatwehave
ρ|t=0=ρ0,∂tρ|t=0=ρ1.
WechooseBi=(1−Δn−1)si/2,withexponentssi>0asin[26,Section4.1]according
totheCases1,2and3.UsingLemma1.3.8andarguingasin[26]onecanshowthat
ρ∈Eρ,µ(R+×Rn−1)asdesired.
(II)Inthegeneralcase,wedescribeΓbyafinitecollectionofcharts(Ui,ϕi)anda
correspondingpartitionofunityψiforΓ,anddenotebyΦithepush-forwardwithrespect
toϕi,i.e.,Φiρ0=ρ0◦ϕi−1.Wefurthertakecut-offfunctionsφi∈Cc∞(Rn−1)with
φi≡1onsuppΦiψi,suppφi⊂ϕi(Ui).
ItfollowsfromLemmaA.4.1thatΦi(ψiρ0)∈Xρ,µ(Rn−1),andalsoΦi(ψiρ1)∈
X∂tρ,µ(Rn−1)incaseκ0>1−µ+1/p.Wedefine
ρ(t)=iΦi−1φi2e−tB0−e−2tB0Φi(ψiρ0)+e−tB1−e−2tB1B1−1Φi(ψiρ1),t>0,


3.2Half-SpaceProblemswithBoundaryConditions

133

whereB0andB1arechosenasaboveaccordingtotheCases1,2and3.Thenρ|t=0=ρ0
and∂tρ|t=0=ρ1,andfurtherρ∈Eρ,µ(R+×Γ)byLemmaA.4.1.
Wenowconsiderthegeneralhalf-spacecasewithconstantcoefficients.TheBanachspace
ofcompatibledataisheregivenby
D=(f,g,u0,ρ0)∈E0,µ×Fµ×Xu,µ×Xρ,µ:forj=1,...,mitholds
Bj(D)u0(∙)+Cj(Dn−1)ρ0(∙)=gj(0,∙)onRn−1ifκj>1−µ+1/p;
g0(0,∙)−B0(D)u0−C0(Dn−1)ρ0∈X∂tρ,µifκ0>1−µ+1/p,
anditisequippedwiththenorm
|(f,g,u0,ρ0)|D=|f|E0,µ+|g|Fµ+|u0|Xu,µ+|ρ0|Xρ,µ
+|g0(0,∙)−B0(D)u0−C0(Dn−1)ρ0|X∂tρ,µ.
Proposition3.2.3.LetEandFbeBanachspacesofclassHT,p∈(1,∞),µ∈(1/p,1],
andassumethat(A,B,C)satisfies(E)and(LS),and,inaddition,forl<2mcondition
(LS∞−),andforl>2mcondition(LS∞+).Letfurtherκj=1−µ+1/pforj=0,...,m.
Thenthereisauniquesolution(u,ρ)∈Eu,µ×Eρ,µof
u+∂tu+A(D)u=f(t,x),x∈R+n,t>0,
ρ+∂tρ+B0(D)u+C0(Dn−1)ρ=g0(t,x),x∈Rn−1,t>0,
Bj(D)u+Cj(Dn−1)ρ=gj(t,x),x∈Rn−1,t>0,j=1,...,m,(3.2.10)
u(0,x)=u0(x),x∈R+n,
ρ(0,x)=ρ0(x),x∈Rn−1,
ifandonlyifthedatasatisfies(f,g,u0,ρ0)∈D.Thecorrespondingsolutionoperator
SH:D→Eu,µ×Eρ,µiscontinuous.
Proof.ThenecessaryconditionsonthedatawerederivedinSection3.1.Ifasolution
operatorexists,thenitiscontinuousduetotheopenmappingtheorem.
Uniquenessofsolutionsof(3.2.10)followsfromLemma3.2.1.Wearegoingtoreduce
theexistenceofasolutionofthefullproblem(3.2.10)totheproblem(3.2.1).Letthe
data(f,g,u0,ρ0)∈Dbegiven.Weextendfandu0toER+nf∈E0,µ(R+×Rn)and
ER+nu0∈Xu,µ(Rn),usingtheextensionoperatorER+nfrom(1.3.3).Proposition2.2.2yields
asolutionv∈Eu,µ(R+×Rn)ofthefull-spaceproblem
v+∂tv+A(D)v=(ER+nf)(t,x),x∈Rn,t>0,
v(0,x)=(ER+nu0)(x),x∈Rn,
whichweusetodefineu:=v|R+n∈Eu,µ.Moreover,thecompatibilityconditionforj=0
imply(3.1.5)andg0|t=0−(ρ0+B0(D)u0+C0(Dn−1)ρ0)∈X∂tρ,µifκ0>1−µ+1/p.

134

MaximalLp,µ-RegularityforBoundaryConditionsofRelaxationType

ItthusfollowsfromLemma3.2.2thatthereisρ∈Eρ,µwithρ|t=0=ρ0and
∂tρ|t=0=g0|t=0−(ρ0+B0(D)u0+C0(Dn−1)ρ0)ifκ0>1−µ+1/p.
Usingthefunctionρ,wedefine
g0:=g0−(ρ+∂tρ+B0(D)u+C0(Dn−1)ρ)∈0F0,µ,
furtherandgj:=gj−(Bj(D)u+Cj(Dn−1)ρ)∈0Fj,µ,j=1,...,m.
Notethattheequalitygj|t=0=0incaseκj>1−µ+1/pfollowsfromthecompatibility
conditionforj=1,...,m.6NowLemma3.2.1yieldsapair(u,ρ)∈0Eu,µ×0Eρ,µsatisfying
u+∂tu+A(D)u=0,x∈R+n,t>0,
ρ+∂tρ+B0(D)u+C0(Dn−1)ρ=g0(t,x),x∈Rn−1,t>0,
Bj(D)u+Cj(Dn−1)ρ=gj(t,x),x∈Rn−1,t>0,j=1,...,m,
u(0,x)=0,x∈R+n,
ρ(0,x)=0,x∈Rn−1.
Thus(u,ρ)=(u+u,ρ+ρ)∈Eu,µ×Eρ,µsolves(3.2.10)byconstruction.
3.2.2TopOrderCoefficientshavingSmallOscillation
Weinvestigatethecaseofoperatorsonahalf-spacewithtopordercoefficientshavingsmall
oscillation,andrestrictourconsiderationstoafiniteinterval
J=(0,T),T>0.
NowtheanisotropicspacesareunderstoodoverJ×R+norJ×Rn−1.Weconsiderthe
eratoroptialdifferenA(t,x,D)=aα(t,x)Dα,x∈R+n,t∈J,
m2|≤α|andtheboundaryoperators
Bj(t,x,D)=bjβ(t,x)trR+nDβ,x∈Rn−1,t∈J,j=0,...,m,
m|≤β|jCj(t,x,Dn−1)=cjγ(t,x)Dnγ−1x∈Rn−1,t∈J,j=0,...,m.
k|≤γ|jThetopordercoefficientsoftheoperatorsareassumedtobeoftheform
aα(t,x)=aα0+aα(t,x),|α|=2m,
6Hereitisnecessarytoexcludethevaluesκj=1−µ+1/p,j=0,...,m,sinceotherwiseitisnot
guaranteedthatgj∈0F0,µ.

3.2Half-SpaceProblemswithBoundaryConditions

135

bjβ(t,x)=bj0β+bjβ(t,x),|β|=mj,j=0,...,m,(3.2.11)
cjγ(t,x)=cj0γ+cjγ(t,x),|γ|=mj,j=0,...,m,
whereaα0,bj0βandcj0γdonotdependontandx.Usingtheseconstantcoefficients,we
defineauxiliaryoperators(A0,B0,C0)by
A0(D):=aα0Dα,(3.2.12)
m=2|α|Bj0(D):=bj0,βtrR+nDβ,Cj0(Dn−1):=cj0γDnγ−1,j=0,...,m.
|β|=mj|γ|=kj
Assuming(SD)and(SB)forthecoefficientsofA−A0andB−B0,thePropositions1.3.16
yield1.3.24andA∈B(Eu,µ(J),E0,µ(J)),B∈B(Eu,µ(J),Fµ(J))
Wenowshowthattheassumption(SC)issufficientfortherequiredcontinuityproperties
.ofCLemma3.2.4.LetFbeofclassHT,letΩ⊂Rnbeadomainwithcompactsmooth
boundaryΓ=∂Ω,orΩ=R+n,andletforalmosteveryt∈Jthedifferentialoperator
C(t,∙,DΓ)=(C0(t,∙,DΓ),...,Cm(t,∙,DΓ)):C∞(Γ;F)→L1(Γ;F×Em)
begiven.Assumethatforj∈{0,...,m}and|γ|≤kjthelocalcoefficientscjgγofCjwith
respecttocoordinatesg:V→Γsatisfycjgγ∈Yjγ(J),whereeither
κj>1−µ+1/p+n2−mp1andYjγ(J)=Fj,µ(J×V;Fj),
holdsitorYjγ(J)=Bsκjjγ,p(J;Lrjγ(V;Fj))∩Lsjγ(J;Br2jγmκ,pj(V;Fj))
withnumbersrjγ,sjγ∈[p,∞)asin(SC).7Thenfor|γ|<kjthereisasmallnumberδ>0
withgγg|cjγDn−1ρ|Fj,µ(J×V)|cjγ|Yjγ(J)|ρ|Wp,µ1+κ0−δ(J;Lp(V;F))∩Lp,µ(J;Wpl+2mκ0−δ(V;F)).
Moreover,forthetopordercoefficients,|γ|=kj,itholdscjgγ∈BUC(J×V;Fj)andthere
isasmallδ>0with
gγg|cjγDn−1ρ|Fj,µ(J×V)|cjγ|BUC(J×V;Fj)|ρ|Eρ,µ(J×V)
g+|cjγ|Yjγ(J)|ρ|Wp,µ1+κ0−δ(J;Lp(V;F))∩Lp,µ(J;Wpl+2mκ0−δ(V;F)).
Inparticular,undertheassumptionsof(SC)theoperatorCextendsto
C∈BEρ,µ(J×Γ),Fµ(J×Γ).
Restrictingtoρ∈0Eρ,µ,andassumingthatthecoefficientsaredefinedonalargertime
intervalJ0=(0,T0)forT0>0,theaboveestimates,withJreplacedbyJ0inthenorms
forthecoefficients,holdwithauniformconstantforT≤T0.
7RecallthatF0=B(F),andFj=B(F,E)forj=1,...,m.

136

MaximalLp,µ-RegularityforBoundaryConditionsofRelaxationType

Proof.Forbothcasestheboundednessandthecontinuityofthetopordercoefficients
followsfromProposition1.3.24.Letj∈{0,...,m}andamultiindexγbegiven.Wehave
thatEρ,µ(J×V)→Wp,µ1+κ0(J;Lp(V;F))∩Wκp,µj(J;Hpkj(V;F))∩Lp,µ(J;Wpl+2mκ0(V;F)),
andProposition1.3.2andthefactthatl+2mκ0≥kj+2mκjyieldforalljthat8
κ+kj−|γ|(1+κ−κ)
Eρ,µ(J×V)→Wp,µjkj0j(J;Hp|γ|(V;F))∩Lp,µ(J;Wp2mκj+kj−|γ|(V;F)).
HenceDnγ−1mapscontinuously
κ+kj−|γ|(1+κ−κ)
Eρ,µ(J×V)→Wp,µjkj0j(J;Lp(V;F))∩Lp,µ(J;Wp2mκj+kj−|γ|(V;F)),
withuniformnorminthe0Eρ,µ-case,andthelatterspaceembedsintoFj,µ(J×V).Incase
κj>1−µ+1/p+2n−mp1andYjγ(J)=Fj,µ(J×V;Fj)theassertedestimatesfollowfrom
Lemma1.3.23.IntheothercasewecanapplytheLemmas1.3.21and1.3.22,with
τ=κj+kjk−|γ|(1+κ0−κj),ϑ=2mκj+kj−|γ|.
jForanintervalJ=(0,T)withT>0theBanachspaceofcompatibledataisgivenby
D(J)=(f,g,u0,ρ0)∈E0,µ(J)×Fµ(J)×Xu,µ×Xρ,µ:forj=1,...,mitholds
Bj(0,∙,D)u0+Cj(0,∙,Dn−1)ρ0=gj(0,∙)onRn−1ifκj>1−µ+1/p;
g0(0,∙)−B0(0,∙,D)u0−C0(0,∙,Dn−1)ρ0∈X∂tρ,µifκ0>1−µ+1/p,
andwealsoconsiderthespace
D0(J)=(f,g,u0,ρ0)∈D(J):g∈0Fµ(J).
Wehavethefollowingresult.
Proposition3.2.5.LetEandFbeBanachspacesofclassHT,p∈(1,∞),µ∈(1/p,1],
andassumethat(A0,B0,C0)satisfy(E),(LS),and,inaddition,forl<2mcondition
(LS∞−)andforl>2mcondition(LS∞+).Supposefurtherthatthecoefficientsof(A−
A0,B−B0,C−C0)satisfy(SD),(SB),(SC),andthatκj=1−µ+1/pforj=0,...,m.
ThenthereareatimeT0∈(0,T]andnumberε>0suchthatif
(t,x)∈[0sup,T0]×Rn|aα(t,x)|B(E)<ε,|α|=2m,(3.2.13)
+(t,x)∈[0,Tsup0]×Rn−1|bjβ(t,x)|Ej+|cjγ(t,x)|Fj<ε,|β|=mj,|γ|=kj,j=0,...,m,9
8UsingthedetailedshapeoftheNewtonpolygonaccordingtotheCases1,2and3,hereonecould
result.ersharpaobtain9RecallthatE0=B(E,F),F0=B(F),andfurtherEj=B(E)andFj=B(F,E)forj=1,...,m.

3.2Half-SpaceProblemswithBoundaryConditions

137

thenforallintervalsJ=(0,T)withT≤T0thereisauniquesolution(u,ρ)=
SHsm(f,g,u0,ρ0)∈Eu,µ(J)×Eρ,µ(J)of
∂tu+A(t,x,D)u=f(t,x),x∈R+n,t∈J,
∂tρ+B0(t,x,D)u+C0(t,x,Dn−1)ρ=g0(t,x),x∈Rn−1,t∈J,(3.2.14)
Bj(t,x,D)u+Cj(t,x,Dn−1)ρ=gj(t,x),x∈Rn−1,t∈J,j=1,...,m,
u(0,x)=u0(x),x∈R+n,
ρ(0,x)=ρ0(x),x∈Rn−1,
ifandonlyif(f,g,u0,ρ0)∈D(J).Thecorrespondingsolutionoperator
SHsm:D(J)→Eu,µ(J)×Eρ,µ(J)
iscontinuous.TheoperatornormofSHsmrestrictedtoD0(J)isuniforminT≤T0.
Proof.Theproofiscompletelyanalogoustothestaticcase,Proposition2.3.1.Throughout,
let0<T≤T0≤T.ThenecessitypartandthedependenceofthesolutionoperatorSHsm
onJcanbeobtainedinthesamewayasintheproofofProposition2.3.1.Thusweonly
havetoshowuniquesolvabilityof(3.2.14)forsufficientlysmallT0andε.
For(f,g,u0,ρ)∈D(J)weset
Zu0,ρ0(J):=(u,ρ)∈Eu,µ(J)×Eρ,µ(J):u(0,∙)=u0,ρ(0,∙)=ρ0,
whichisanonemptysetduetotheLemmas1.3.9and3.2.2,andisfurtheraclosedsubspace
ofEu,µ(J)×Eρ,µ(J)byTheorem1.3.6.Forgiven(u,ρ)∈Zu0,ρ0(J)weconsiderthe
problemv+∂tv+A0v=f+(A0−A+1)uinR+n,t∈J,
σ+∂tσ+B00v+C00σ=g0+(B00−B0)u+(C00−C0+1)ρonRn−1,t∈J,(3.2.15)
Bj0v+Cj0σ=gj+(Bj0−Bj)u+(Cj0−Cj)ρonRn−1,t∈J,j=1,...,m,
v(0,∙)=u0inR+n,
σ(0,∙)=ρ0,onRn−1,
where(A0,B0,C0)isdefinedin(3.2.12).Since(u,ρ)∈Zu0,ρ0(J),theright-handsidesof
theboundaryequationsin(3.2.15)belongtoDB0,C0(J),thespaceofcompatibledatafor
(A0,B0,C0).ItcanbeseenasinStepIoftheproofofLemma3.2.1thatforeach(u,ρ)
solutionsof(3.2.15)areuniqueinEu,µ(J)×Eρ,µ(J).Thesolution(v,σ)=S(u,ρ)∈
Eu,µ(J)×Eρ,µ(J)of(3.2.15)isgivenby
S(u,ρ):=SHEJ(f+(A0−A+1)u),EJ(g+(B0−B)u+(C0−C+1)ρ),u0,ρ0|J.(3.2.16)
HereSH:DB0,C0(R+)→Eu,µ(R+)×Eρ,µ(R+)isthesolutionoperatorfor(3.2.15)onthe
half-line,whichisgivenbyProposition3.2.3anddefinedonDB0,C0(R+).Further,EJis
theextensionoperatorfromJtoR+,givenbyLemma1.1.5.
UsingtheLemmas1.3.21,1.3.22and1.3.23,onecanshowasintheproofofProposition
2.3.1thatShasauniquefixedpoint(u,ρ)∈Zu0,ρ0(J),providedT0andεaresufficiently

138

MaximalLp,µ-RegularityforBoundaryConditionsofRelaxationType

small.NotethattoshowthecontractionpropertyofSthecontinuityofSHisonlyemployed
forvanishinginitialvaluesandboundarydatafrom0Fµ.Byconstruction,thisfixedpoint
istheuniquesolutionof(3.2.14)inEu,µ(J)×Eρ,µ(J).

3.3TheGeneralCaseonaDomain
Wecannowprovethemainresultofthischapter,Theorem3.1.4,employingalocalization
procedureanalogouslytotheproofofTheorem2.1.4.Foranoutlineoftheproofwerefer
totheendofSection2.1.
Tosetthescene,letEandFbeBanachspacesofclassHT,letJ=(0,T)beafinite
interval,andletΩ⊂RnbeadomainwithcompactsmoothboundaryΓ=∂Ω.Weconsider
problemthe∂tu+A(t,x,D)u=f(t,x),x∈Ω,t∈J,
∂tρ+B0(t,x,D)u+C0(t,x,DΓ)ρ=g0(t,x),x∈Γ,t∈J,
Bj(t,x,D)u+Cj(t,x,DΓ)ρ=gj(t,x),x∈Γ,t∈J,j=1,...,m,(3.3.1)
u(0,x)=u0(x),x∈Ω,
ρ(0,x)=ρ0(x),x∈Γ,
wherethedifferentialoperatorsAandB=(B0,...,Bm)aregivenby
A(t,x,D)=aα(t,x)Dα,t∈J,x∈Ω,
m2|≤α|Bj(t,x,D)=bjβ(t,x)trΩDβ,t∈J,x∈Γ,mj∈{0,...,2m−1},
m|≤β|jandwheretheoperatorsC=(C0,...,Cm)areinlocalcoordinatesggivenby
Cjg(t,x,DΓ)=cjgγ(t,x)Dnγ−1,t∈J,kj∈N0,j=0,...,m.
k|≤γ|jAssumingthatthecoefficientsaα,bjβ,andcjgγsatisfy(SD),(SB)and(SC),itfollowsfrom
thePropositions1.3.16,1.3.24andLemma3.2.4that
A∈B(Eu,µ,E0,µ),B∈B(Eu,µ,Fµ),C∈B(Eρ,µ,Fµ).
Forthetopordercoefficients,itisassumedinresp.followsfrom(SD),(SB)and(SC)that
aα∈BUC(J×Ω;B(E)),|α|=2m,(3.3.2)
bjβ∈BUC(J×Γ;Ej),|β|=mj,cjgγ∈BUC(J×Γ;Fj),|γ|=kj,j=0,...,m,
whereEjandFjaredefinedin(SC).TheBanachspaceofcompatibledataisgivenby
D=(f,g,u0,ρ0)∈E0,µ×Fµ×Xu,µ×Xρ,µ:forj=1,...,mitholds
Bj(0,∙,D)u0+Cj(0,∙,DΓ)ρ0=gj(0,∙)onΓifκj>1−µ+1/p;
g0(0,∙)−B0(0,∙,D)u0−C0(0,∙,DΓ)ρ0∈X∂tρ,µifκ0>1−µ+1/p,
andwealsoconsiderthespace
D0=(f,g,u0,ρ0)∈D:g∈0Fµ.

3.3TheGeneralCaseonaDomain

139

3.1.4.TheoremofofProTheargumentsoftheStepsIandIIoftheproofofTheorem2.1.4concerningthenecessary
conditions,continuityofthesolutionoperator,realsolutionsandcausalitycarryoverto
thepresentsituation.Wethusonlyhavetoshowthatfor(f,g,u0,ρ0)∈Dthereexistsa
uniquesolution(u,ρ)∈Eu,µ×Eρ,µof(3.3.1),wherewemayassumethatTissufficiently
small.(I)Welocalizeinspace,anduseProposition2.3.2totreatthelocalproblemswithout
boundaryconditions,andProposition3.2.5forthelocalproblemswithboundarycondi-
tions.AsintheproofofTheorem2.1.4wetakeafinitenumberofpoints
xi∈Ω,i=1,...,NH,
togetherwithx0:=∞ifΩisunbounded,andcorrespondingopenneighbourhoodsUi⊂Rn
ofthesepoints(whereU0=∅ifΩisbounded)satisfying
NHΩ⊂Ui,Ui∩Γ=∅,i=0,...,NF,Ui∩Γ=∅,i=NF+1,...,NH.
=0iFurther,theboundaryΓ⊂iN=HNF+1Uiisdescribedbycharts(Ui,ϕi),i=NF+1,...,NH,
havingpropertiesasin(2.4.9),i.e.,
ϕi(xi)=0,ϕi(Ui)=B2ri(0),ϕi(xi)=Oν(xi),
ϕi(Ui∩Ω)⊂R+n,ϕ(Ui∩Γ)⊂Rn−1,Ui=ϕi−1Bri(0).
Herewehaveri>0,andOν(xi)istheorthogonalmatrixrotatingν(xi)to(0,...,0,−1)∈
RN,whichwehavefixedfortheformulationoftheLopatinskii-Shapiroconditions.
(II)Fori=0,...,NFwedefineextendedcoefficientsaiαonJ×Rn,|α|≤2m,suchthat
aiα|J×Ui=aα,
asin(2.4.4),(2.4.6)and(2.4.7),respectively.Thisyieldsoperators
Ai(t,x,D):=aiα(t,x)Dα
m2|≤α|whichsatisfy(E),andwhosecoefficientssatisfy(SD),respectively.Ifthediametersofthe
Uiaresufficientlysmall,byProposition2.3.2thereisforallsufficientlysmallT=|J|a
continuoussolutionoperator
SFsm,i:E0,µ(J×Rn)×Xu,µ(Rn)→Eu,µ(J×Rn)
problemfull-spacethefor∂tv+Ai(t,x,D)v=fi(t,x),x∈Rn,t∈J,(3.3.3)
v(0,x)=u0i(x),x∈Rn.

(3.3.3)

140

MaximalLp,µ-RegularityforBoundaryConditionsofRelaxationType

(III)Fori=NF+1,...,NHwedenotebyΦithepush-forwardoperatorwithrespectto
ϕi,i.e.,Φiu=u◦ϕi−1,anddefinethetransformedoperatorsAΦiandBΦiby
AΦi(t,x,D):=ΦiA(t,∙,D)Φi−1(x),t∈J,x∈R+n∩Bri(0),
BΦi(t,x,D):=ΦiB(t,∙,D)Φi−1(x),t∈J,x∈Rn−1∩Bri(0).
DenotingbygithelocalparametrizationofΓcorrespondingto(Ui,ϕi),wefurtherdefine
thelocalizedoperatorCgi=(C0gi,...,Cmgi)by
Cjgi(t,x,Dn−1):=cjgγi(t,x)Dγn−1,t∈J,x∈Rn−1∩Bri(0),j=0,...,m.
k|≤γ|jHerecjgγidenotethecoefficientsfromthelocalrepresentationofCjwithrespecttogi.
ThecoefficientsofAΦiareextendedtocoefficientsaiαonJ×R+nasin(2.4.6)and(2.4.7).
Moreover,thecoefficientsofBΦiandCgiareextendedtocoefficientsbjiβandcjiγonJ×Rn−1
asin(2.4.11)and(2.4.12).
Theseextendedcoefficientsyieldoperators(Ai,Bi,Ci)onthehalf-space.Wedefinetop
orderconstantcoefficientoperators(Ai,0,Bi,0,Ci,0)by
Ai,0(D):=aαi,0Dα,aαi,0:=aiα(0,xi),
m=2|α|Bji,0(D):=bji,β0trR+nDβ,bji,β0:=bjiβ(0,xi),j=0,...,m,
|β|=mj
Cji,0(Dn−1):=cji,γ0Dnγ−1,cji,γ0:=cjgγi(0,xi),j=0,...,m.
|β|=kj
ItfollowsfromLemmaA.1.2thatforξ∈Rn−1itholds
Ai,0(ξ,Dy)=A(0,xi,OνT(xi)(ξ,Dy)),Bi,0(ξ,Dy)=B(0,xi,OνT(xi)(ξ,Dy)).
Hence,byassumption,wehavethatAi,0satisfies(E),and(Ai,0,Bi,0,Ci,0)satisfiesthe
Lopatinskii-ShapiroconditionsonR+n.Thecoefficientsof(Ai−Ai,0,Bi−Bi,0,Ci−Ci,0)
satisfy(SD),(SB)and(SC)byconstruction.Givenε>0,ifT,riandthediameterofUi
aresufficientlysmall,thenthetopordercoefficientsof(Ai−Ai,0,Bi−Bi,0,Ci−Ci,0)have
oscillation.εTherefore(Ai,Bi,Ci)satisfiesforalli=NF+1,...,NHtheassumptionsofProposition
3.2.5,andtherearecontinuoussolutionoperators
SHsm,i:DBi,Ci(J)→Eu,µ(J×R+n)×Eρ,µ(J×R+n)
problemsthefor∂tv+Ai(t,x,D)v=fi(t,x),x∈R+n,t∈J,
∂tσ+B0i(t,x,D)v+C0i(t,x,Dn−1)σ=g0i(t,x),x∈Rn−1,t∈J,(3.3.4)
Bji(t,x,D)v+Cji(t,x,Dn−1)σ=gji(t,x),x∈Rn−1,t∈J,j=1,...,m,
v(0,x)=u0i(x),x∈R+n,
σ(0,x)=ρ0i(x),x∈Rn−1,

3.3TheGeneralCaseonaDomain

141

providedT,riandthediameterofUiaresufficientlysmall,respectively.HereDBi,Ci(J)
denotesthespaceofcompatibledatafor(Ai,Bi,Ci).
(IV)Denotingby{ψi}i=0,...,NHthepartitionofunityforΩsubordinatetothecover
iN=0HUi,asconstructedinStepVIIIoftheproofofTheorem2.1.4,wetakeφi∈Cc∞(Rn),
i=0,...,NH,with
φi≡1onsuppψi,suppφi⊂Ui.
AsintheproofofTheorem2.1.4itthenholdsthatif(u,ρ)∈Eu,µ×Eρ,µsolves(3.3.1)
withdata(f,g,u0,ρ0)∈D,then(u,ρ)isafixedpointofthemapGf,g,u0,ρ0,definedby
NFGf,g,u0,ρ0(u,ρ):=φiSFsm,i(fi,u0i;u)|Ui,0
=0iNH+φiΦi−1(SHsm,i(fi,gi,u0i,ρ0i;u,ρ)|R+n∩Bri(0)),
+1N=iFspacehBanactheonZu0,ρ0(J):={(u,ρ)∈Eu,µ×Eρ,µ:u(0,∙)=u0,ρ(0,∙)=ρ0},
whichisnontrivialduetotheLemmas1.3.9and3.2.2.Herefori=0,...,NFwehaveset
fi:=ψif+[A,ψi]u,u0i:=ψiu0,
andfurther,fori=NF+1,...,NH,
fi:=Φi(ψif+[A,ψi]u),gi=Φi(ψig+[B,ψi]u+[C,ψi]ρ),
u0i:=Φi(ψiu0),ρ0i:=Φi(ψiρ0),
andthenotationsSFsm,i(fi,u0i;u)andSHsm,i(fi,gi,u0i,ρ0i;u,ρ)indicatethatfi,gi,u0iand
ρ0iaredefinedwithrespecttothefunctionsuandρ,respectively.Moreover,[∙,∙]denotes
thecommutatorbracket.
Usingthatthecommutatorsareoflowerorder,asinStepIXoftheproofofTheorem2.1.4
onecanshowthatforall(f,g,u0,ρ0)∈D(J)themapGf,g,u0,ρ0hasauniquefixedpoint
inZu0,ρ0(J),prosmvided,iTissufficientlysmall.Noteherethattherequiredcompatibility
conditionsforSHattheboundaryaretriviallysatisfied,sincegi|t=0,ifitexists,is
independentof(u,ρ)∈Zu0,ρ0(J).Thisyieldsafixedpointmap
Q:D(J)→Zu0,ρ0(J),Q(f,g,u0,ρ0)=Gf,g,u0,ρ0(Q(f,g,u0,ρ0)),
withthepropertythat
Q:{(f,g,0,0)∈D0(J)}→Z0,0(J)
iscontinuouswithoperatornormuniforminTsmallerthanagivenlength.
(V)AsintheStepsXandXIoftheproofofTheorem2.1.4,forgiven(f,g,u0,ρ0)∈
D(J)onecannowfindtheappropriatedata(f,g,u0,ρ0)∈D(J)suchthat(u,ρ)=

142

MaximalLp,µ-RegularityforBoundaryConditionsofRelaxationType

Q(f,g,u0,ρ0)solves(3.3.1)withdata(f,g,u0,ρ0)bysolvingonemorefixedpointequa-
tion.WritingQ=Q(f,g,u0,ρ0),hereoneobtainsforthedynamicequationonthe
ndaryoub(B0,∂t+C0)∙Q=φiΦi−1Φi(B0,∂t+C0)Φi−1∙SHsm,if,i,g,i,u0i;Q|R+n∩Bri(0)
NH
+1N=iFNH+[(B0,C0),φi]∙Φi−1SHsm,if,i,g,i,u0i;Q|R+n∩Bri(0)
+1N=iFNH=g0+φi[(B,C),ψi]Q+K20(f,g),
+1N=iFwherethecorrectiontermK20(f,g)isgivenby
NHK20(f,g):=[(B0,C0),φi]∙Φi−1SHsm,if,i,g,i,u0i;Q|R+n∩Bri(0).
+1N=iFHereallthetermscontainingSFsm,ivanish,sincethefunctionsφivanishonΓfori=
0,...,NF.Moreover,as{ψi}isapartitionofunityforΓandφi≡1onsuppψiitholds
thatNHφi[(B,C),ψi]Q=[(B,C),1]Q=0.
+1N=iFSimilarly,incaseκ0>1−µ+1/p,duetoQ(f,g,u0,ρ0)|t=0=(u0,ρ0)wehave
NHK20(f,g)|t=0=[(B0(0,∙,D),C0(0,∙,DΓ)),φi]∙ψi(u0,ρ0)=0,
+1N=iFwhichyieldsthatK20mapsinto0F0,µ(J).DefiningthecorrectiontermsK1forthedynamic
equationinΩand(K21,...,K2m)forthestaticboundaryequationsasinStepXoftheproof
ofTheorem2.1.4,respectively,andsettingK2=(K20,...,K2m),theappropriate(f,g)is
ofsolutionthe(f,g)+(K1,K2)(f,g)=(f,g).
ThisequationcanberewrittentoafixedpointproblemonE0,µ(J)×0Fµ(J),andcanbe
solvedviathecontractionprincipleasinStepXIoftheproofofTheorem2.1.4.

4Chapter

AttractorsinStrongerNormsfor
ConditionsBoundaryRobin

nductiotroIn4.1Inthischapterweareconcernedwiththelong-timebehaviourofsemilinearandquasilinear
reaction-diffusionsystemsinseparateddivergenceformwithRobinboundaryconditions.
Fortheunknownu=u(t,x)∈RN,whereN∈N,weconsidertheproblem1
∂tu−∂i(aij(u)∂ju)=f(u)inΩ,t>0,
aij(u)νi∂ju=g(u)onΓ,t>0,(4.1.1)
u(0,∙)=u0inΩ.
HereΩ⊂RnisaboundeddomainwithsmoothboundaryΓ=∂Ω,n≥2,andtheouter
normalunitfieldonΓisdenotedbyν=(ν1,...νn).Itisassumedthat(4.1.1)isofseparated
i.e.,m,forergencedivaij(u)=a(u)αij∈B(RN),i,j∈{1,...,n},
wherea:RN→B(RN)andwheretheαij∈Rareconstants,i,j∈{1,...,n}.Weimpose
thefollowingstructuralconditionsonthesecoefficients.
(αij)i,j=1,...,nissymmetricanduniformlypositivedefinite,
σ(a(ζ))⊂C+={Rez>0},ζ∈RN.(4.1.2)
Wefurtherassumethroughoutthataandthereactiontermsf,g:RN→RNaresmooth.
Notethattheaboveassumptionsallowtorewritetheboundaryconditionintotheequiv-
formtalenαijνi∂ju=a−1(u)g(u)onΓ,t>0.
Thusforg=0andαij=δij,theKroneckersymbol,oneobtainshomogeneousNeumann
conditions.daryounb1∂i(aijW(eu)∂useju)msumustbconevenreadtion,asPi.e.,ni,jit=1∂isi(aij(uundersto)∂juo).dthatonesumsoverdoubleindices.Forinstance,

144

AttractorsinStrongerNormsforRobinBoundaryConditions

Parabolicsystemsoftype(4.1.1)modelmanydifferentphenomenainphysics,chemistry
andbiology.Fora≡idandαij=δijoneobtainsareactiondiffusionsystemwithnonlin-
earboundaryconditions.AlsotheKeller-SegelmodelforchemotaxisandtheShigesada-
Kawasaki-Teramotocross-diffusionmodelforpopulationdynamicscanbecastintheform
4.5.onSecticf.(4.1.1),

Localwell-posednessinascaleofSlobodetskiispacesforproblemsoftype(4.1.1)iswell
knownandwasestablishedbyAmann[4,5,6].Beingprecise,Theorem14.4andCorollary
14.7of[6]yieldthefollowing.
Theorem4.1.1.Letp∈(n,∞)ands∈(n/p,1+1/p).Assumethat(4.1.2)holds,and
furtherthatg(u)=g(u)uwithasmoothfunctiong:RN→B(RN).Thenforu0∈
Wps(Ω,RN)thereisauniquemaximalsolution
u(∙,u0)∈C[0,t+(u0));Wps(Ω,RN)∩C∞(0,t+(u0))×Ω,RN
of(4.1.1),wheret+(u0)>0denotesthemaximalexistencetime.Thesolutionmapu0→
u(∙,u0)definesacompactlocalsemiflowonWps(Ω,RN).
WerefertothebeginningofSection4.3forthenotionofacompactlocalsemiflow.Fora
generalboundaryreactiontermgthesystem(4.1.1)isstilllocallywell-posedintheabove
scale,butthensmoothnessofthesolutionsisamoredelicateissue,ingeneral,cf.[6].
Criteriaforglobalexistenceofsolutions,t+(u0)=+∞,werealsoestablishedbyAmann
[4].Roughlyspeaking,anaprioriHölderboundissufficientforasolutiontoexistglobally,
asTheorem15.3of[6]shows.Inmanyspecialcases,liketriangularsystems,itsufficesto
findanL∞-bound[6,Theorem15.4],orevenweakerbounds.

Onceglobalexistenceisestablished,oneisinterestedinthelong-timebehaviourofsolu-
tions,especiallyintheconvergencetoequilibriaortheexistenceofaglobalattractor.Let
p∈(1,∞),s≥0andMpsbeasubsetofWps(Ω,RN).AnonemptycompactsetA⊂Mps
iscalledaglobalattractorof(4.1.1)if(4.1.1)generatesasemiflowofglobalsolutionsin
Mps,ifAisinvariantunderthesemiflow(u(t,A)⊂Aforallt≥0)andifitattractsevery
boundedsubsetMofMps,i.e.,itholds
dH(u(t,M),A):=u0∈usup(t,M)v0inf∈A|u0−v0|Wps(Ω,RN)→0ast→+∞,
withrespecttotheHausdorffdistancedH.Inthissensetheflowontheattractor,ifitexists,
determinesthelong-timebehaviourofsolutions.Itisfurtherknownthatanattractoris
unique,andthatinMpsitholds
A=theunionoftheω-limitsetsofallboundedsets
=theunionofallboundedcompleteorbits
=themaximalboundedinvariantset
=theminimalboundedsetthatattractsallboundedsets.

ductionIntro4.1

145

Wereferto[16,49,63]formoreinformations.NotethatAcontainsinparticularallequi-
libria,allperiodicsolutionsandallheteroclinicorbitsof(4.1.1).IfAhasfiniteHausdorff
dimension,thentheglobaldynamicsof(4.1.1)reducetoafinitedimensionalprocess,which
isofessentiallylesscomplexitythantheoriginal,infinitedimensionalone.Itistherefore
desireabletohaveanattractorinanormasstrongaspossible,sincealthoughthesolutions
containedinAmightbesmooth,thesolutionsapproachtheattractoronlywithrespect
tothenorminMpswhereAwasestablished.
AssumethatanattractorAexistsinWps∗forsomes∗∈(0,2)inthesemilinearcase,i.e.,
ifadoesnotdependonu,andforlinearboundaryconditions.Itisthenaconsequenceof
thevariationofconstantsformulathatthesolutionsapproachAintheWps-normforall
s∈(s∗,2)andthatAisindependentofs,cf.[16,Section4.3].Thusoneautomaticallyhas
convergencetoAinstrongernorms.
Itisthepurposeofthischaptertoshowthatacorrespondingresultforattractorsin
strongernormsisvalidalsointhequasilinearcasewithnonlinearboundaryconditions,i.e.,
forthefullproblem(4.1.1).Thekeytothesemiflowinhighernormsandthesubstitutefor
thevariationofconstantsformulaisthemaximalLp,µ-regularityresultgivenbyTheorem
2.1.4.

Letusconsidertheresultsindetail.WefirstshowinSection4.3that(4.1.1)generatesa
compactlocalsemiflowinthescaleofnonlinearphasespaces
Mps:=u0∈Wps(Ω,RN):aij(u0)νi∂ju0=g(u0)onΓ,
wherep∈(n+2,∞)ands∈(1+n/p,2−2/p].Thisrangeofregularityisnotcovered
byAmann’stheory.Foreacht∈(0,t+(u0))thesolutionsbelongtotheweightedmaximal
assclyregularitEu,µ(0,τ):=W1p,µ(0,τ;Lp(Ω,RN))∩Lp,µ(0,τ;Wp2(Ω,RN))
wheretheweightµ∈(1/p,1]issuchthats=2(µ−1/p).Ourresultisbasedonthe
regularitypropertiesofthenonlinearsuperpositionoperatorscorrespondingtofandg,
whichareinvestigatedinSection4.2,andonmaximalLp,µ-regularityforthelinearized
problem,Theorem2.1.4.Ourargumentscanalsobeusedtoestablishalocalsemiflowin
ascaleofnonlinearphasespacesasaboveformuchmoregeneralsystemsthan(4.1.1),as
treatedin[65]fors=2−2/pwithoutweights.
InSection4.4wethenusemaximalLp,µ-regularityandtheinherentsmoothingeffectofthe
weightedspacestoshowthatif(4.1.1)hasanabsorbantsetinaHölderspaceCα(Ω,RN),
α>0,i.e.,thereisC>0suchthateachsolutionsatisfies
t→limt+(usup0)|u(t,u0)|Cα(Ω,RN)≤C,(4.1.3)
then(4.1.1)hasaglobalattractorinthephasespaceMps.Sinces>1+n/p,thisin
particularyieldstheconvergencetotheattractorintheC1-norm,andasaresultAalso

146

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determinesthelong-timebehaviourofthespatialgradientofsolutionswithrespecttothe
sup-norm.Wealsoconsiderspecialcaseswheretheaboveresultremainstrueifonereplacesthe
Cα-normin(4.1.3)byaweakernorm,likesemilinearproblems,cross-diffusionmodelsand
equations.singleInSection4.5weillustratetheseresultsinobtaininganattractorinstrongernormsfor
semilinearreaction-diffusionsystemswithnonlinearboundaryconditions,forachemotaxis
modelwithvolumefillingeffect,andforapopulationmodelwithcross-diffusion.

Besidesthemaximalregularityclass
Eu,µ(J)=W1p,µ(J;Lp(Ω,RN))∩Lp,µ(J;Wp2(Ω,RN)),
throughoutthischapterweworkwiththeweightedspace
E0,µ(J):=Lp,µ(J;Lp(Ω,RN)),
whereJ=(0,T)isafiniteinterval,p∈(1,∞)andµ∈(1/p,1].Sincetheboundary
operatorin(4.1.1)isoforder1,thespacefortheboundaryequationis
Fµ(J):=W1p,µ/2−1/2p(J;Lp(Γ,RN))∩Lp,µ(J;Wp1−1/p(Γ,RN)).
ItfollowsfromTheorem1.3.6that
Eu,µ(J)→C(J;B2(p,pµ−1/p)(Ω,RN)),
andthereforeSobolev’sembeddingsyield
Eu,µ(J)→C(J;C1(Ω,RN))if2(µ−1/p)>1+n/p.(4.1.4)
holdsit,SimilarlyFµ(J)→C(J;B2(p,pµ−1/p)−1−1/p(Γ,RN))if2(µ−1/p)>1+1/p,
sothatwehave
Fµ(J)→C(J;C(Γ,RN))if2(µ−1/p)>1+n/p.(4.1.5)
Restrictingto0Eu,µ-and0Fµ-spaces,theconstantsfortheaboveembeddingsareindepen-
dentoftheunderlyingintervalJ.

4.2SuperpositionOperators
Forourpurposesitisconvenienttorewrite(4.1.1)intotheabstractform
∂tu+A(u)=0inΩ,t>0,
B(u)=0onΓ,t>0,
u(0,∙)=u0inΩ,

4.2SuperpositionOperators

147

wherethenonlineardifferentialoperatorsAandBareforu∈Eu,µ(J)givenby
A(u):=−∂i(aij(u)∂ju)+f(u),B(u):=αijνitrΩ∂ju−a−1(trΩu)g(trΩu).
ThepurposeofthissectionistoinvestigatetheregularitypropertiesofAandB.Westart
withsomeuniformestimatesfornonlinearfunctions.
Lemma4.2.1.Letψ:Rm→RMbesmoothform,M∈N,andletBR(0)⊂Rmbea
fixedclosedballaroundtheoriginwithradiusR.Inthesequelwedenotebyεacontinuous
functionε:[0,∞)→[0,∞)withε(0)=0.
a)Thereisafunctionεasabovewith
|ψ(ξ+η)−ψ(ξ)−ψ(ξ)η|≤ε(|η|)|η|forallξ,η∈BR(0).
b)Defineφ:Rm×Rm→RMbyφ(ξ,η):=ψ(ξ+η)−ψ(ξ)−ψ(ξ)η.Thenthereisε
asabovewith
|φ(ξ2,η2)−φ(ξ1,η1)|≤ε(|η1|+|η2|)|η2−η1|+|η1||ξ2−ξ1|
forallξ1,ξ2,η1,η2∈BR(0).
c)Defineϕ:Rm×Rm→RMbyϕ(ξ,η):=ψ(ξ+η)−ψ(ξ).Thenthereareεasabove
andaconstantC>0with
|ϕ(ξ2,η2)−ϕ(ξ1,η1)|≤ε(|η2|)|ξ2−ξ1|+C|η2−η1|
forallξ1,ξ2,η1,η2∈BR(0).
Proof.(I)Sinceψissmooth,forξ0∈BR(0)thereisafunctionεξ0asabovesuchthat
|ψ(ξ0+η)−ψ(ξ0)−ψ(ξ0)η|<εξ0(|η|)|η|,η∈BR(0).(4.2.1)
Bycontinuityandcompactness,(4.2.1)holdstruewithξ0replacedbyξforallξinasmall
neighbourhoodofξ0.Bycompactnesswefindfinitelymanyξisuchthattheseneighbour-
hoodscoverBR(0),withcorrespondingfunctionsεi.Nowε:=maxiεisatisfiestheasserted
inequalityina)forallξ,η∈BR(0).
(II)Toshowb),weestimatewiththemeanvaluetheorem
|φ(ξ2,η2)−φ(ξ1,η1)|≤s∈[0sup,1]|∂ξφ(sξ2+(1−s)ξ1,η1)||ξ2−ξ1|
+sup|∂ηφ(ξ2,sη2+(1−s)η1)||η2−η1|.(4.2.2)
1],[0∈sForξ,η∈BR(0),η=0,theterms
|∂ξφ(ξ,η)|/|η|=|ψ(ξ+η)−ψ(ξ)−ψ(ξ)η|/|η|
and

|∂ηφ(ξ,η)|=|ψ(ξ+η)−ψ(ξ)|

148

AttractorsinStrongerNormsforRobinBoundaryConditions

tendtozeroas|η|→0uniformlyinξ,bya)andtheuniformcontinuityofψonBR(0).
Applyingthisto(4.2.2)showsb).Assertionc)isshowninasimilarway.
Wenowconsiderthepropertiesofthemap
A(u)=−∂i(aij(u)∂ju)+f(u),u∈Eu,µ(J).
Lemma4.2.2.LetJ=(0,T)befinite,andletp∈(n+2,∞)andµ∈(1/p,1]besuch
that2(µ−1/p)>1+n/p.ThenA∈C1(Eu,µ(J),E0,µ(J)),andforu∈Eu,µ(J)wehave
A(u)h=−∂i(aij(u)∂jh+aij(u)∂juh)+f(u)h,h∈Eu,µ(J).
Moreover,letT0,R>0begiven.Thenthereisacontinuousfunctionε:[0,∞)→[0,∞)
withε(0)=0suchthatforT≤T0itholds
|A(u+h)−A(u)−A(u)h|E0,µ(J)≤ε(|h|Eu,µ(J))|h|Eu,µ(J)(4.2.3)
forallu,h∈Eu,µ(J)with
h(0,∙)=0,|u|C(J;C1(Ω,RN)),|u|Eu,µ(J),|h|Eu,µ(J)≤R.(4.2.4)
Proof.Throughoutweset
|∙|0,∞:=|∙|C(J;C(Ω,RN)),|∙|1,∞:=|∙|C(J;C1(Ω,RN)).
(I)Itiseasytoseethattheestimate
|vw|Wp1(Ω,RN)≤|v|C(Ω,RN×N)|w|Wp1(Ω,RN)+|v|Wp1(Ω,RN×N)|w|C(Ω,RN)(4.2.5)
isvalidforallv∈Wp1(Ω,RN×N)andw∈Wp1(Ω,RN),providedp>n.Weusethisfact
andtheembedding(4.1.4)toestimateforu,h∈Eu,µ(J)2
|A(u+h)−A(u)−A(u)h|E0,µ(J)
|f(u+h)−f(u)−f(u)h|E0,µ(J)+|a(u)∂jhh|Lp,µ(J;Wp1(Ω,RN))
+|aij(u+h)−aij(u)−aij(u)h∂j(u+h)|Lp,µ(J;Wp1(Ω,RN))
|f(u+h)−f(u)−f(u)h|0,∞+|aij(u)|1,∞|h|E2u,µ(J)(4.2.6)
+|aij(u+h)−aij(u)−aij(u)h|1,∞(|u|Eu,µ(J)+|h|Eu,µ(J)).
Notethatforh(0)=0theseestimatesareuniforminT≤T0.Forthefirstsummandin
(4.2.6)wehave,usingLemma4.2.1andagain(4.1.4),
|f(u+h)−f(u)−f(u)h|0,∞≤ε(|h|0,∞)|h|0,∞≤ε(|h|Eu,µ(J))|h|Eu,µ(J).(4.2.7)
Incase(4.2.4),theimagesofuandharecontainedinacompactsubsetofRN,which
yieldsthatεisuniforminT≤T0andR.Further,thesecondsummandin(4.2.6)may
beestimatedbyε(|h|Eu,µ(J))|h|Eu,µ(J),whereεisagainuniformfor(4.2.4).Forthethird
2Itisunderstoodthatonetakesthemaximumoversingleindices.

4.2SuperpositionOperators

149

summandwehavethatthesecondfactorthereisbounded,anditisuniformlyboundedin
case(4.2.4).Forthefirstfactorthere,wedenotebythegradientonRnandwritethe
|∙|1,∞normas|∙|0,∞+|∙|0,∞.The|∙|0,∞partmaybeestimatedasin(4.2.7).Forthe
|∙|0,∞partwehave,estimatingagainasin(4.2.7)andusing(4.1.4),
|(aij(u+h)−aij(u)−aij(u)h)|0,∞
≤|aij(u)hh|0,∞+|aij(u+h)−aij(u)−aij(u)h|0,∞(|u|1,∞+|h|1,∞)
|aij(u)|0,∞|h|E2u,µ(J)+ε(|h|0,∞)|h|0,∞
≤ε(|h|Eu,µ(J))|h|Eu,µ(J),
withtheasserteddependenceonTincase(4.2.4).Thisshowstheuniformestimate(4.2.3),
andfurtherthatAisdifferentiableineachu∈Eu,µ(J)withderivativeA(u).
(II)ItremainstoshowthatA:Eu,µ(J)→B(Eu,µ(J),E0,µ(J))iscontinuous.Forthiswe
takeu,v,h∈Eu,µ(J)with|h|Eu,µ(J)≤1.Thenitfollowsfrom(4.2.5)and(4.1.4)that
|(A(u)−A(v))h|E0,µ(J)
≤|(f(u)−f(v))h|E0,µ(J)+|(aij(u)−aij(v))∂jh|Lp,µ(J;Wp1(Ω,RN))
+|(aij(u)∂ju−aij(v)∂jv)h|Lp,µ(J;Wp1(Ω,RN))
|f(u)−f(v)|0,∞+|aij(u)−aij(v)|1,∞+|aij(u)∂ju−aij(v)∂jv|0,∞
+|aij(u)(∂ju−∂jv)|Lp,µ(J;Wp1(Ω,RN×N))+|(aij(u)−aij(v))∂jv)|Lp,µ(J;Wp1(Ω,RN×N))
|f(u)−f(v)|0,∞+|aij(u)−aij(v)|1,∞+|aij(u)∂ju−aij(v)∂jv|0,∞
+|aij(u)(∂ju−∂jv)|1,∞+|(aij(u)−aij(v))∂jv)|1,∞,
andthisconvergestozeroasu→vinEu,µ(J)dueto(4.1.4).
Wenextinvestigatetheregularityofsuperpositionoperatorsontheboundary.Theestimate
ina)isusefulforlowvaluesofqandµ.
Lemma4.2.3.ForafiniteintervalJ=(0,T)andasmoothfunctiong:RN→RNthe
true.sholdwingfolloa)Letq∈(1,∞),µ∈(1/q,1],andκ,τ∈(0,1).Then
|g(u)|Wq,µκ,τ(J×Γ,RN)ζ∈supBu|g(ζ)||u|Wq,µκ,τ(J×Γ,RN)+|g(u)|C(J×Γ,RN)
forallu∈Wq,µκ,τ(J×Γ,RN)∩C(J×Γ,RN),whereBuisaballwithu(J×Γ)⊂Bu.
b)Letnowp∈(n+2,∞)andµ∈(1/p,1]satisfy2(µ−1/p)>1+n/p.Thenforthe
superpositionoperatorG,givenbyG(u):=g(trΩu),wehave
G∈C1(Eu,µ(J),Fµ(J)),G(u)=g(trΩu)trΩ.
c)Inthesituationofb),letT0,R>0begiven.Thenthereisacontinuousfunction
ε:[0,∞)→[0,∞)withε(0)=0suchthatforT≤T0itholds
|g(u+h)−g(u)−g(u)h|0Fµ(J)≤ε(|h|Eu,µ(J))|h|Eu,µ(J)

150

AttractorsinStrongerNormsforRobinBoundaryConditions

forallu,h∈Eu,µ(J)satisfying
h(0,∙)=0,|u|C(J;C1(Γ,RN)),|u|Eu,µ(J),|u(0,∙)|Wp2(µ−1/p)(Ω,RN),|h|Eu,µ(J)≤R.
(4.2.8)Proof.(I)Toshowa),takeu∈Wq,µκ,τ(J×Γ,RN)∩C(J×Γ,RN).Thenitholds
|g(u)|Lq,µ(J;Lq(Γ,RN))|g(u)|C(J×Γ,RN).
FortheintrinsicseminormoftheweightedSlobodetskiispaces,givenbyProposition1.1.13,
weestimatewiththemeanvaluetheorem
qqTstq(1−µ)
[g(u)]Wqκ,µ(J;Lq(Γ,RN))=1+κq|g(u(s,x))−g(u(t,x))|dσ(x)dtds
00Γ(s−t)
qq≤sup|g(ζ)|[u]Wqκ,µ(J;Lq(Γ,RN)).
B∈ζu(II)Toestimate|g(u)|Lq,µ(J;Wqτ(Γ,RN))wedescribeΓbyafinitecollectionofcharts(Ui,ϕi),
chooseapartitionofunity{ψi}forΓsubordinatetoiUiandsetWi:=suppψi⊂Ui.
Thenforalmosteveryt∈Jwehave
|g(u(t,∙))|Wqτ(Γ,RN)|ψi(ϕi−1)g(u(t,ϕi−1))|Wqτ(ϕi(Wi),RN).
iForeachiitholds,asabove
|ψi(ϕi−1)g(u(t,ϕi−1))|Lq(ϕi(Wi),RN)|g(u)|C(J×Γ,RN).
FortheseminormcorrespondingtoWqτ(ϕi(Wi),RN),cf.(A.4.2),weestimate
q1−1−[ψi(ϕi)g(u(t,ϕi))]Wqτ(ϕi(Wi),RN)
|ψi(ϕ−1(x))g(u(t,ϕ−1(x)))−ψi(ϕ−1(y))g(u(t,ϕ−1(y)))|q
=iiiidxdy
ϕi(Wi)2|x−y|n−1+τq
q|u(t,ϕi−1(x))−u(t,ϕi−1(y))|qq
sup|g(ζ)||x−y|n−1+τqdxdy+|g(u)|C(J×Γ,RN)
ζ∈Buϕi(Wi)2
qqqsup|g(ζ)|[u(t,∙)]Wqτ(Γ,RN)+|g(u)|C(J×Γ,RN),
B∈ζuwherewehaveusedLemmaA.4.1inthelastline.Summingoveri,usingtheaboveestimates
andtakingtheLq,µ-normleadsto
|g(u)|Lq,µ(J;Wqτ(Γ,RN))sup|g(ξ)||u|Lq,µ(J;Wqτ(Γ,RN))+|g(u)|C(J×Γ,RN),
B∈ξua).implieshwhic(III)WenextshowdifferentiabilityofG.Foru∈Eu,µ(J)itfollowsfrom2(µ−1/p)>
1+n/p,p>nandTheorem1.3.6that
p,µtrΩu∈W1−1/2p,2−1/p(J×Γ,RN)→C(J×Γ,RN)∩Lp,µ(J;C1(Γ,RN)).
Hencea)impliesg(trΩu)∈Fµ(J)∩C(J×Γ,RN),andLemma1.3.23yieldsg(trΩu)trΩ∈
B(Eu,µ(J),Fµ(J)).ToshowthedifferentiabilityofGatu∈Eu,µ(J),takeh∈Eu,µ(J).

4.2SuperpositionOperators

151

ArguingasinStepIoftheproofofLemma4.2.2weobtainthatthereisε:[0,∞)→[0,∞)
withε(0)=0,whichisuniforminRfor(4.2.8),suchthat3
|g(u(t,∙)+h(t,∙))−g(u(t,∙))−g(u(t,∙))h(t,∙)|C1(Γ,RN)
≤ε(|h(t,∙)|C1(Γ,RN))|h(t,∙)|C1(Γ,RN)
isvalidforalmostallt∈J.TakingtheLp,µ-normandusingC1(Γ,RN)→Wp1−1/p(Γ,RN)
and(4.1.4)weobtain
|g(u+h)−g(u)−g(u)h|Lp,µ(J;Wp1−1/p(Γ,RN))≤ε(|h|C(J;C1(Γ,RN)))|h|C(J;C1(Γ,RN))
ε(|h|Eu,µ(J))|h|Eu,µ(J).
ObservethattheseestimatesarealwaysuniforminT≤T0andRif(4.2.8)holds.
(IV)FortheintrinsicseminormofW1p,µ/2−1/2p(J,Lp(Γ,RN)),whichisgivenbyProposition
setew1.1.13,Ξ(t,x):=g(u(t,x)+h(t,x))−g(u(t,x))−g(u(t,x))h(t,x)
andestimate,usingLemma4.2.1,
[g(u+h)−g(u)−g(u)h]pW1p,µ/2−1/2p(J;Lp(Γ,RN))
Tstp(1−µ)p
=00Γ(s−t)1+(1/2−1/2p)p|Ξ(s,x)−Ξ(t,x)|dσ(x)dtds
ppp≤ε(|h|C(J×Γ,RN))[h]Wκp,µ(J;Lp(Γ,RN))+|h|C(J×Γ,RN)[u]Wκp,µ(J;Lp(Γ,RN))
ε(|h|Eu,µ(J))|h|Eu,µ(J).(4.2.9)
NotethattheseestimatesarealsovalidonR+.TogetherwiththeestimatesofStep
III,weobtainthatGisdifferentiableateachu∈Eu,µ(J).Butsincewehaveused
theintrinsicnormoverJ,(4.2.9)doesnotyieldanestimateuniformlyinTinthe
0W1p,µ/2−1/2p(J;Lp(Γ,RN))-casefor(4.2.8)(seealsothediscussioninRemark1.1.15).
Toovercomethisobstacle,letu,h∈Eu,µ(J)beasin(4.2.8).DuetoLemma1.3.9thereis
u∗∈Eu,µ(R+)with
u∗(0,∙)=u(0,∙),|u∗|Eu,µ(R+)|u(0,∙)|Wp2(µ−1/p)(Ω,RN).
Usingthisfunctionwedefine
u:=EJ0(u−u∗)+u∗∈Eu,µ(R+),h:=EJ0h∈0Eu,µ(R+),
whereEJ0istheextensionoperatorfromLemma1.1.5whosenormisindependentofT.
thateObserv|u|BC([0,∞)×Ω,RN)|u|Eu,µ(R+)R+|u(0,∙)|Wp2(µ−1/p)(Ω,RN),
3InthesequelweneglectthespatialtracetrΩforbetterreadability.

152

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and,duetoh(0)=0,
|h|BC([0,∞)×Ω)|h|Eu,µ(R+)|h|Eu,µ(J)≤R,
wheretheseestimatesareindependentofT.Therefore,ifweapplyLemma4.2.1togwith
argumentsfromtheimagesu(R+×Ω)andh(R+×Ω),thentheresultingfunctionsε
willdependonamultipleofR+|u(0,∙)|Wp2(µ−1/p)(Ω,RN),butnotonT≤T0.Thus,using
Proposition1.1.11and(4.2.9)onthehalf-line,wemayestimate
|g(u+h)−g(u)−g(u)h|0W1p,µ/2−1/2p(J;Lp(Γ,RN))
≤|g(u+h)−g(u)−g(u)h|0W1p,µ/2−1/2p(R+;Lp(Γ,RN))
|g(u+h)−g(u)−g(u)h|W1p,µ/2−1/2p(R+;Lp(Γ,RN))
≤ε(|h|BC([0,∞)×Γ,RN))[h]W1p,µ/2−1/2p(R+;Lp(Γ,RN))
+|h|BC([0,∞)×Γ,RN)[u]W1p,µ/2−1/2p(R+;Lp(Γ,RN))
ε(|h|Eu,µ(J))|h|Eu,µ(J)R+|u(0)|Wp2(µ−1/p)(Ω,RN),
wherethefunctionεisuniforminT≤T0andR.Thisshowsc).
(V)Forb)itislefttoshowthatG:Eu,µ(J)→B(Eu,µ(J),Fµ(J))iscontinuous.Tothis
endtakeu,v,h∈Eu,µ(J)with|h|Eu,µ(J)≤1.Thenweestimate,usingLemma1.3.23,
|(g(u)−g(v))h|Fµ(J)|g(u)−g(v)|C(J×Γ,RN×N)+|g(u)−g(v)|W1p,µ/2−1/2p(J;Lp(Γ,RN×N))
+|g(u)−g(v)|Lp,µ(J;Wp1−1/p(Γ,RN×N)).
Asu→vinEu,µ(J),thefirstsummandconvergestozero.Forthesecondsummandwe
estimateand4.2.1Lemmause[g(u)−g(v)]W1p,µ/2−1/2p(J;Lp(Γ,RN×N))ε(|u−v|C(J×Γ,RN))[v]W1p,µ/2−1/2p(J;Lp(Γ,RN))
+|u−v|W1p,µ/2−1/2p(J;Lp(Γ,RN)).
Heretheright-handsideconvergestozeroasu→v.UsingC1(Γ,RN)→Wp1−1/p(Γ,RN),
weobtainthesameforthethirdsummand.Thusb)isfinallyproved.
Forthenonlinearboundaryoperator
B(u)=αijνitrΩ∂ju−a−1(trΩu)g(trΩu)
theabovelemmayieldsthefollowing.
Lemma4.2.4.LetJ=(0,T)befinite,andletp∈(n+2,∞)andµ∈(1/p,1]besuch
that2(µ−1/p)>1+n/p.ThenB∈C1(Eu,µ(J),Fµ(J)),withderivative
B(u)=αijνitrΩ∂j−a−1g(trΩu)trΩ,u∈Eu,µ(J).
Further,letT0,R>0begiven.Thenthereisacontinuousfunctionε:[0,∞)→[0,∞)
withε(0)=0,suchthatforT≤T0itholds
|B(u+h)−B(u)−B(u)h|0Fµ(J)≤ε(|h|Eu,µ(J))|h|Eu,µ(J)
forallu,h∈Eu,µ(J)asin(4.2.8).

4.3TheLocalSemiflow

153

4.3TheLocalSemiflow
Forp∈(1,∞)ands∈(1+n/p,2−2/p]recallthenonlinearphasespace
Mps=u0∈Wps(Ω,RN):B(u0)=0,
whichisequippedwiththemetricfromWps(Ω,RN).Wesaythat(4.1.1)generatesacom-
pactlocalsemiflowofEu,µ-solutionsonMpsifthefollowingthreeconditionsaresatisfied.
1.Forallu0∈Mpsthereist+(u0)>0suchthat(4.1.1)hasauniquemaximal
solutionu(∙,u0)∈C[0,t+(u0));Wps(Ω,RN)whichbelongstoEu,µ(0,τ)forall
τ∈(0,t+(u0)).
2.Forallu0∈Mpsandτ∈(0,t+(u0))thereisr>0suchthatt+(v0)>τforall
v0∈Br(u0)∩Mps,andthemapu(τ,∙):Br(u0)∩Msp→Mpsiscontinuous.
3.IfforaboundedsetM⊂Mpsthereisτ>0suchthatt+(v0)>τforallv0∈M,
thenu(τ,M)isrelativelycompactinMps.
Toverifythefirstconditionfor(4.1.1)weconsiderthelinearinitial-boundaryvalueproblem
associatedto(A(u),B(u)),andshowthatitenjoysmaximalLp,µ-regularityforeach
u∈Eu,µ(J).
Lemma4.3.1.LetJ=(0,T)beafiniteinterval,andletp∈(n+2,∞)andµ∈(1/p,1]
thathsucebs:=2(µ−1/p)>1+n/p.
Assumefurtherthat(4.1.2)isvalid,andletthefunctionu∈Eu,µ(J)begiven.Denoteby
Du(J):=(f,g,v0)∈E0,µ(J)×Fµ(J)×Wps(Ω,RN):B(u(0,∙))v0=g0onΓ
thespaceofcompatibledatawithrespectto(A(u),B(u)).Thenthereexistsabounded
linearsolutionoperatorL:Du(J)→Eu,µ(J)for
∂tv+A(u(t,x))v=f(t,x),x∈Ω,t∈J,
B(u(t,x))v=g(t,x),x∈Γ,t∈J,
v(0,x)=v0(x),x∈Ω.
GivenT0>0,theoperatornormofLrestrictedto
Du0(J):=(f,g,v0)∈Du(J):g∈0Fµ(J)
isuniforminT≤T0.
Proof.Wecheckthat(A(u),B(u))satisfiestheassumptionsofTheorem2.1.4.Since
u∈C(J;C1(Ω,RN))by(4.1.4),thetopordercoefficientsofA(u)belongtoBUC(J×
Ω,RN×N).ThelowerordercoefficientsbelongtoE0,µ(J;RN×N).Moreover,Lemma4.2.3
impliesthatthecoefficientsofB(u)belongtoFµ(J;RN×N).Sincethecondition1/2−

154

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1/2p>1−µ+1/p+n2−p1isequivalentto2(µ−1/p)>1+n/p,weobtainthatthe
coefficientssatisfythefirstconditionsin(SD)and(SB),respectively.
Itremainstochecktheellipticityconditions.Tothisendconsiderthepair(A(u),B),given
ybA(u)v:=∂iaij(u)∂jv),Bv:=αijνitrΩ∂jv,v∈Eu,µ(J).
Itisshownin[5,Theorem4.4]that(4.1.2)implies(E)and(LS)for(A(ζ),B),ζ∈RN.
Sincetheseconditionsareindependentofthelowerordertermsitfollowsthat(A(u),B(u))
satisfies(E)and(LS)aswell.ThusalltheassumptionsofTheorem2.1.4aresatisfied,and
ws.folloassertiontheNowwecanprovelocalexistenceanduniquenessforsolutionsof(4.1.1).Ourproofisbased
onmaximalLp,µ-regularityforthelinearizedproblemandthecontractionprinciple,and
follows[90](seealso[59,65]).
Proposition4.3.2.Letp∈(n+2,∞)andµ∈(1/p,1]besuchthats=2(µ−1/p)>
1+n/p,andassumethat(4.1.2)isvalid.Thenforeachinitialvalueu0∈Wps(Ω,RN)with
aij(u0)νi∂ju0=g(u0)onΓ(4.3.1)
thesystem(4.1.1)hasauniquemaximalsolution
u(∙,u0)∈C[0,t+(u0));Wps(Ω,RN),
whichbelongstoEu,µ(0,τ)forallτ∈(0,t+(u0)).
Proof.Werewrite(4.1.1)intotheequivalentform
∂tu+A(u)=0inΩ,t>0,
B(u)=0onΓ,t>0,(4.3.2)
u(0,∙)=u0inΩ,
whereAandBweredefinedinthebeginningoftheprevioussection.Notethatthe
condition(4.3.1)onu0isequivalenttoB(u0)=0.Throughouttheproofwefixu∗∈
Eu,µ(R+)withu∗(0,∙)=u0,whichexistsbyLemma1.3.9.
(I)Weconsiderthelinearproblem
∂tw+A(u∗)w=A(u∗)u∗−A(u∗)inΩ,t>0,
B(u∗)w=B(u∗)u∗−B(u∗)onΓ,t>0,(4.3.3)
w(0,∙)=u0inΩ.
DuetotheLemmas4.2.2and4.2.4itholds
A(u∗)u∗−A(u∗)∈E0,µ(0,1),B(u∗)u∗−B(u∗)∈Fµ(0,1),
andsinceB(u0)=0thecompatibilitycondition
B(u0)u0=B(u0)u0−B(u0)onΓ

4.3TheLocalSemiflow

155

istriviallysatisfied.ThusLemma4.3.1yieldsauniquesolutionw∗∈Eu,µ(0,1)of(4.3.3).
Usingw∗,wedefineforσ,τ∈(0,1]
Σ(σ,τ):=u∈Eu,µ(0,τ):|u−w∗|Eu,µ≤σ,u(0,∙)=u0.
ThesetΣ(σ,τ)isclosedinEu,µ(0,τ).Moreover,(4.1.4)implies
|u|C([0,τ];C1(Ω,RN)),|u(0,∙)|Wp2(µ−1/p)(Ω,RN),|u|Eu,µ(0,τ)1+|w∗|Eu,µ(0,1),(4.3.4)
uniformlyinu∈Σ(σ,τ)andσ,τ∈(0,1].Foru∈Σ(σ,τ)wenextconsider
∂tw+A(u∗)w=A(u∗)u−A(u)inΩ×(0,τ),
B(u∗)w=B(u∗)u−B(u)onΓ×(0,τ),(4.3.5)
w(0,∙)=u0inΩ.
Asabove,forallτ∈(0,1]thereisauniquesolutionw=S(u)∈Eu,µ(0,τ)of(4.3.5)due
toLemma4.3.1.Thisdefinesamap
S:Σ(σ,τ)→Eu,µ(0,τ).
Observethatu∈Σ(σ,τ)solves(4.3.2)on(0,τ)ifandonlyifitisafixedpointofS
inΣ(σ,τ).Sinceforgivenσeachsolutionof(4.3.2)inEu,µ(0,τ)belongstoΣ(σ,τ)for
sufficientlysmallτ,ourtaskisthustoshowthatthemapShasauniquefixedpoint
inΣ(σ,τ),providedthatσandτaresufficientlysmall.Tothisendweusethecon-
tractionprinciple.Theexistenceofamaximalexistencetimeandamaximalsolution
inC([0,t+(u0));Wps(Ω,RN))thenfollowsfromstandardarguments.
(II)FirstweshowthatSisaselfmaponΣ(σ,τ)forsmallσandτ.Foru∈Σ(σ,τ)the
differencez=S(u)−w∗solves
∂tz+A(u∗)z=A(u∗)−A(u)−A(u∗)(u∗−u)inΩ×(0,τ),
B(u∗)z=B(u∗)−B(u)−B(u∗)(u∗−u)onΓ×(0,τ),
z(0,∙)=0inΩ.
Notethattheright-handsideoftheboundaryequationbelongsto0Fµ(0,τ).Thusby
Lemma4.3.1thereisaconstantC0,independentofτ,suchthat
|S(u)−w∗|Eu,µ(0,τ)≤C0|A(u∗)−A(u)−A(u∗)(u∗−u)|E0,µ(0,τ)
+|B(u∗)−B(u)−B(u∗)(u∗−u)|0Fµ(0,τ).(4.3.6)
Asaboveitholds
|u∗−u|Eu,µ(0,τ)σ+|w∗−u∗|Eu,µ(0,1),(4.3.7)
uniformlyinu∈Σ(σ,τ)andτ∈(0,1].UsingthisfacttogetherwithLemma4.2.2we
obtainthatthefirstsummandin(4.3.6)maybeestimatedby
|A(u∗)−A(u)−A(u∗)(u∗−u)|E0,µ(0,τ)≤ε(|u∗−u|Eu,µ(0,τ))|u∗−u|Eu,µ(0,τ)
≤ε(|u∗−w∗|Eu,µ(0,τ)+σ)(|u∗−w∗|Eu,µ(0,τ)+σ),

156

AttractorsinStrongerNormsforRobinBoundaryConditions

whereε:[0,∞)→[0,∞)isacontinuousfunctionwithε(0)=0,whichisindependentof
σ,τ∈(0,1].Wefirstchooseσwithε(2σ)≤1/4C0andthenτsuchthatthefixedfunctions
u∗andw∗satisfy|u∗−w∗|Eu,µ(0,τ)≤σ.Thenweobtain
|A(u∗)−A(u)−A(u∗)(u∗−u)|E0,µ(0,τ)≤σ/2C0.
Similarly,using(4.3.7)andLemma4.2.4weobtainforthesecondsummandin(4.3.6)that
|B(u∗)−B(u)−B(u∗)(u∗−u)|0Fµ(0,τ)≤σ/2C0
aswell,providedσandτaresufficientlysmall.ThisshowsthatSisaselfmappingon
Σ(σ,τ)ifσandτareappropriatelychosen.
(III)WeshowthatSisastrictcontractiononΣ(σ,τ).Foru,v∈Σ(σ,τ)wehaveasabove
|S(u)−S(v)|Eu,µ(0,τ)≤C0|A(u∗)u−A(u)−A(u∗)v+A(v)|E0,µ(0,τ)(4.3.8)
+|B(u∗)u−B(u)−B(u∗)v+B(v)|0Fµ(0,τ).
Using(4.3.4),(4.3.7)andLemma4.2.2,weestimatethefirstsummandin(4.3.8)by
|A(u∗)u−A(u)−A(u∗)v+A(v)|E0,µ(0,τ)
≤|A(v)−A(u)−A(u)(v−u)|E0,µ(0,τ)+|A(u∗)−A(u)(u−v)|E0,µ(0,τ)
≤ε(|v−u|Eu,µ(0,τ))+|A(u∗)−A(u)|B(Eu,µ(0,τ),E0,µ(0,τ))|v−u|Eu,µ(0,τ)
≤ε(2σ)+ε(σ+|w∗−u∗|Eu,µ(0,τ))|v−u|Eu,µ(0,τ),
whereεisafunctionasabove,independentofσ,τ∈(0,1].Thusifσandτaresufficiently
obtainewsmall|A(u∗)u−A(u)−A(u∗)v+A(v)|E0,µ(0,τ)≤1/4C0|v−u|Eu,µ(0,τ).
UsingLemma4.2.4,inthesamewayweobtainforthesecondsummandin(4.3.8)that
|B(u∗)u−B(u)−B(u∗)v+B(v)|0Fµ(0,τ)≤1/4C0|v−u|Eu,µ(0,τ).
ThisshowsthatSisastrictcontractiononΣ(σ,τ)ifσandτaresufficientlysmall.
Beforewetreatthecontinuousdependenceontheinitialvaluesweneedanotherprepara-
toryresultontheboundaryoperatorB.
Lemma4.3.3.Letp∈(n+2,∞)andµ∈(1/p,1]satisfy2(µ−1/p)>1+n/p.Thenwe
evhaB∈C1Wp2(µ−1/p)(Ω,RN),Wp2(µ−1/p)−1−1/p(Γ,RN),
eativderivwithB(u0)=αijνitrΩ∂j−a−1g(trΩu0)trΩforu0∈Wp2(µ−1/p)(Ω,RN).
Further,if(4.1.2)isvalid,thenforeachu0themapB(u0)issurjectivewithbounded
linearright-inverse.

4.3TheLocalSemiflow

157

Proof.ByLemma1.3.9,forallu0∈Wp2(µ−1/p)(Ω,RN)thereisu∗∈Eu,µ(0,1)with
u∗(0,∙)=u0,whichdependssmoothlyonu0.ItthusfollowsfromB(u0)=tr0B(u∗),
Lemma4.2.4andTheorem1.3.6thatBisC1,withderivativeasasserted.
Fortheright-inverseofB(u0)weintendtouseProposition2.5.1.Considertheoperators
A:=αij∂i∂j,B:=αijνitrΩ∂j.
Then(4.1.2)and[5,Theorem4.4]yieldthat(A,B)satisfies(E)and(LS),andthusalso
(A,B(u0))satisfies(E)and(LS).FortheregularityofthecoefficientsofB(u0),onecan
showasinStepIIoftheproofofLemma4.2.3that
a−1g(trΩu0)∈Wp2(µ−1/p)−1/p(Γ,RN×N)→Wp2(κ−(1−µ+1/p))(Γ,RN×N),
whereκ=1/2−1/2p.ThusB(u0)satisfiestheassumptionsofProposition2.5.1,andthe
existenceofacontinuousright-inversefollows.Itisclearthatforreal-valuedu0,aandg
thisright-inversemapsintoaspaceofreal-valuedfunctions.
Thefollowingresultonthecontinuousdependenceofsolutionsontheinitialdataisbased
onacombinationofmaximalLp,µ-regularityandtheimplicitfunctiontheorem.Wefollow
theproofof[65,Theorem14].
Proposition4.3.4.InthesettingofProposition4.3.2,letu=u(∙,u0)bethemaximal
solutionof(4.1.1)withinitialvalueu0∈Mps.Thenforallτ∈(0,t+(u0))thereisaball
Br(u0)inWps(Ω,RN),r>0,andacontinuousmap
Φ:Br(u0)∩Mps→Eu,µ(0,τ),Φ(u0)=u,
suchthatΦ(v0)isthesolutionof(4.1.1)on(0,τ)withinitialvaluev0∈Br(u0)∩Mps.
Proof.(I)Takep∈(n+2,∞)andµ∈(1/p,1]withs=2(µ−1/p),suchthatu∈
Eu,µ(0,τ).Weconsiderthelinearproblem
∂tz+A(u(t,x))z=f(t,x),x∈Ω,t∈(0,τ),
B(u(t,x))z=g(t,x),x∈Γ,t∈(0,τ),(4.3.9)
z(0,x)=w0(x),x∈Ω,
ybdenoteandS:Du(0,τ)→Eu,µ(0,τ)
theboundedlinearsolutionoperatorcorrespondingto(4.3.9)fromLemma4.3.1.Wehave
thatv∈Eu,µ(0,τ)solves(4.1.1)(andtherewrittenproblem(4.3.2))withinitialvalue
v0∈Mpsifandonlyif
v=u+SF(v−u),G(v−u),v0−u0,(4.3.10)
wherethenonlinearfunctionsFandGaregivenby
F(w):=−A(u+w)−A(u)−A(u)w,G(w):=−B(u+w)−B(u)−B(u)w.


158

AttractorsinStrongerNormsforRobinBoundaryConditions

DuetotheLemmas4.2.2and4.2.4itholds
F∈C1Eu,µ(0,τ),E0,µ(0,τ),G∈C1Eu,µ(0,τ),Fµ(0,τ).
(II)WedefinethetangentialspaceofMpsatu0by
Tu0Mps:=z0∈Wps(Ω,RN):B(u0)z0=0.
ThisisthekerneloftheboundedlinearoperatorB(u0)inWps(Ω,RN),andthusaBanach
space.Wefurtherconsiderthenonlinearmap
F:Tu0Mps×Eu,µ(0,τ)→Eu,µ(0,τ),
ybdefinedF(z0,w):=w−SF(w),G(w),z0+Nstr0G(w).
HereNs∈BWps−1−1/p(Γ,RN),Wps(Ω,RN)denotesthecontinuousright-inverseofB(u0)
fromLemma4.3.3,andtr0isthetemporaltraceatt=0,i.e.,tr0w=w(0,∙).ThemapF
iswelldefined,sincedueto
B(u0)z0+Nstr0G(w)=tr0G(w)
onlycompatibledataareinsertedintoS.ItfurtherholdsF(0,0)=0andthatFis
continuouslydifferentiable.ThederivativeofFwithrespecttothesecondargumentat
(z0,w)=(0,0)isgivenby
∂2F(0,0)
=SA(u+w)−A(u),B(u+w)−B(u),Nstr0(B(u+w)−B(u))|w=0=id,
andisthereforeinvertible.ThuswecansolvethenonlinearequationF(z0,w)=0locally
around(0,0)uniquelybyw=Φ∗(z0)withaC1-functionΦ∗:Br(0)→Eu,µ(0,τ),where
Br(0)⊂Tu0Mpsandr>0issmall.
(III)Nowletv0∈Mpsbegiven,anddefine
z0:=id−NsB(u0)(v0−u0)∈Tu0Mps.
Bycontinuityofid−NsB(u0),ifv0isclosetou0inMpsthenthenormofz0inWps(Ω,RN)
issmall,suchthatw=Φ∗(z0)∈Eu,µ(J)iswell-definedandsatisfies
w=SF(w),G(w),v0−u0−Ns(B(u0)(v0−u0)−tr0G(w)).
Duetotr0G(w)=−B(u0+w(0,∙))+B(u0)(w(0,∙)),thecontinuityofNs,B(v0)=0and
yield4.3.3Lemma|w(0,∙)−(v0−u0)|Wps(Ω,RN)
=|NsB(u0+w(0,∙))−B(u0)(w(0,∙)−(v0−u0))|Wps(Ω,RN)
|B(u0+w(0,∙))−B(v0)−B(v0)(w(0,∙)−(v0−u0))|Wps−1−1/p(Ω,RN)
+|B(v0)−B(u0)(w(0,∙)−(v0−u0))|Wps−1−1/p(Ω,RN)
≤ε|w(0,∙)−(v0−u0)|Wps(Ω,RN)+|v0−u0|Wps(Ω,RN)|w(0,∙)−(v0−u0)|Wps(Ω,RN),


4.3TheLocalSemiflow

159

withε(0)=0.SinceΦ∗iscontinuousandsatisfiesΦ∗(0)=0,ifv0tendstou0then
|w(0,∙)|Wps(Ω,RN)tendstozero.Thusforv0sufficientlyclosetou0theaboveinequalityis
onlypossibleifw(0,∙)=v0−u0.Thisimpliesthatthefunctionv=u+w∈Eu,µ(0,τ)
solves(4.3.10),andtherefore(4.1.1)withinitialvaluev0.Now
Φ(v0):=u+Φ∗(id−NsB(u0))(v0−u0)
istheassertedcontinuoussolutionmapfor(4.1.1)onBr(u0)∩Mps.
Theaboveresultinparticularshowsthat(4.1.1)satisfiesalsothesecondconditionfora
compactlocalsemiflow.Wenowprovetherequiredcompactnesspropertyofthesolution
map,employingtheinherentsmoothingeffectoftheLp,µ-spaces.Ourargumentsaresimilar
tothoseinSection3oftherecentpaper[59].
Proposition4.3.5.InthesettingofProposition4.3.2,lettheboundedsetM⊂Mpsand
τ>0satisfyt+(v0)>τforallv0∈M.Thenu(τ,M)isrelativelycompactinMps.
Proof.(I)ItfollowsfromthecompactnessoftheembeddingWp1(Ω,RN)→Lp(Ω,RN),cf.
[1,Theorem6.3],andtheinterpolationresultin[7,SectionI.2.7]thatfors∗∈(1+n/p,s)
eddingbemtheWps(Ω,RN)→Wps∗(Ω,RN)
iscompact.ThereforeMisrelativelycompactinWps∗(Ω,RN).Takeµ∗∈(1/p,1]with
s∗=2(µ∗−1/p).DuetoProposition4.3.4,foreachv0∈MthereisaballBr(v0)in
Wps∗(Ω,RN)andacontinuousmap
Φ:Br(v0)∩Mps∗→Eu,µ∗(0,τ)
suchthatw=Φ(w0)∈Eu,µ∗(0,τ)solves(4.1.1)withinitialvaluew0∈Br(v0)∩Mps∗.
manyballsBkandmapsΦkwiththeabovepropertysuchthatkBkcoversM.
ThisyieldsanopencoverofMinWps∗(Ω,RN),andthus,bycompactness,therearefinitely
(II)EachΦkmapstherelativelycompactsetBk∩McontinuouslyintoEu,µ∗(0,τ),with
Φk(w0)=u(∙,w0)|(0,τ),w0∈Bk∩M.
traceoraltemptheSincetrτ:Eu,µ∗(0,τ)→Wp2−2/p(Ω,RN),trτw=w(τ,∙),
iscontinuous,weobtainthat
u(τ,M)=trτ◦Φk(Bk∩M)
kisrelativelycompactinWps(Ω,RN),asacontinuousimageofarelativelycompactset.
Wesummarizetheaboveconsiderationstothemainresultofthissection.

160

AttractorsinStrongerNormsforRobinBoundaryConditions

Theorem4.3.6.Letp∈(n+2,∞),s∈(1+n/p,2−2/p]andµ∈(1/p,1]satisfy
s=2(µ−1/p),andassumethat(4.1.2)holdstrue.Thenthesystem
∂tu−∂i(aij(u)∂ju)=f(u)inΩ,t>0,
aij(u)νi∂ju=g(u)onΓ,t>0,
u(0,∙)=u0inΩ,
generatesacompactlocalsemiflowofEu,µ-solutionsonthephasespaceMps.
Remark4.3.7.Themethodsinthissectionareindependentoftheconcreteformofthe
nonlinearoperatorsAandB,aslongastheyareC1andTheorem2.1.4isapplicableto
thecorrespondinglinearizedproblem.Thusacompactlocalsemiflowinascaleofnonlin-
earphasespacescanbeobtainformuchmoregeneralparabolicsystemswithnonlinear
boundaryconditions,astreatedin[65],forinstance.

4.4GlobalAttractorsinStrongerNorms
Wenowfixp∈(n+2,2−∞2)/pandinvestigatethelong-timebehaviourofsolutionsof(4.1.1)for
initialvaluesfromMp.UsingthefullstrengthofmaximalLp,µ-regularityweestimate
solutionsof(4.1.1)atalatertimeinastrongnormbythesolutionatanearliertimein
aweakernorm.Thisbuildsthebridgefromlowertohigherregularity,andisthekeyto
globalattractorsinstrongernorms.
Lemma4.4.1.Letu0∈Mp2−2/p,anddenotebyu(∙,u0)themaximalsolutionof(4.1.1).
Letq∈(1,p],µ∈(1/q,1],set
σ:=2(µ−1/q)∈(0,2−2/q],
andassumethatσ∈/{1,1+1/q}.Letfurtherτ>0,0<T1<T2<t+(u0)andτ=T2−T1.
Thenforα>0thereisaconstantC=Cτ,α,|u(∙,u0)|C([T1,T2],Cα(Ω,RN))with
|u(T2,u0)|Wq2−2/q(Ω,RN)≤C1+|u(T1,u0)|Wqσ(Ω,RN).(4.4.1)
Inthesemilinearcase,i.e.,if(aij)doesnotdependonu,onemaytakeα=0.
Proof.ThroughoutwesetJ:=(0,τ).ThespacesE0,µ,Eu,µandFµmustnowbeunder-
stoodwithrespecttoq,e.g.,E0,µ(J)=Lq,µ(J;Lq(Ω,RN)).
(I)Definethefunctionv∈Wp1(J;Lp(Ω,RN))∩Lp(J;Wp2(Ω;RN))by
v(t):=u(t+T1,u0),t∈J.
Sincetheweightonlyhasaneffectatt=0,wehave

|u(T2,u0)|Wq2−2/q(Ω,RN)=|v(τ)|Wq2−2/q(Ω,RN)|v|Eu,µ(J),

(4.4.2)

4.4GlobalAttractorsinStrongerNorms

161

independenceonτ.Moreover,thefunctionvsolvesthenonautonomous,inhomogeneous
problemlinear∂tw−aij(v)∂i∂jw=aij(v)∂iv∂jv+f(v)inΩ,t∈J,
αijνi∂jw=a−1(v)g(v)onΓ,t∈J,
w(0,∙)=u(T1,u0)inΩ.
ItfollowsfromTheorem2.1.4,localizationargumentssimilartothoseintheproofof
Proposition2.3.1andcompactnessthatthereisaconstantC,whichisuniformin
|u|C([T1,T2]×Ω,RN)andτ,suchthat
1−|v|Eu,µ(J)≤C|aij(v)∂iv∂jv|E0,µ(J)+|f(v)|E0,µ(J)+|a(v)g(v)|Fµ(J)+|u(T1,u0)|Wqσ(Ω,RN).
(4.4.3)(II)UsingHölder’sinequalityweestimateforthefirstsummandin(4.4.3)
qq|aij(v)∂iv∂jv|E0,µ(J)|u|C([T1,T2]×Ω,RN)|∂iv∂jv|E0,µ(J)
qq(1−µ)2q
≤||∂iv|L2q(Ω,RN)|∂jv|L2q(Ω,RN)|Lq,µ(J)≤t|v(t)|W21q(Ω,RN)dt.
JBytheGagliardo-Nirenberginequality(PropositionA.6.2)wehaveforallt∈Jthat
qqq2|v(t)|W1(Ω,RN)|v(t)|Wqϑ(Ω,RN)|v(t)|Wrτ(Ω,RN)
q2forr∈(1,∞)andϑ,τ>0,provided1−2q<2τ−r+2ϑ−q.Forgivenαitholds
n1n1n
Cα(Ω,RN)→Wrτ(Ω,RN)forτ∈(0,α)andr∈(1,∞).Thusifϑ<2issufficientlyclose
to2andrislargeweobtainfromtheinterpolationinequalityandYoung’sinequality
qqq2|v(t)|W21q(Ω,RN)|v(t)|Wqϑ(Ω,RN)|v(t)|Cα(Ω,RN)
q|u|C([T1,T2];Cα(Ω,RN))ε|v(t)|Wq2(Ω,RN)+Cε|u|C([T1,T2]×Ω,RN),
whereε>0maybechosenarbitrarysmall.Wethereforehave
|aij(v)∂iv∂jv|E0,µ(J)|u|C([T1,T2];Cα(Ω,RN))ε|v|Eu,µ(J)+Cε.
Observethatthistermdoesnotoccurinthesemilinearcase.
(III)Forthesecondsummandin(4.4.3)itiseasilyseenthat
|f(v)|E0,µ(J)|f(u)|C([T1,T2]×Ω,RN).
Forthethirdsummand,Lemma4.2.3,theinterpolationinequalityandYoung’sinequality
yield|g(v)|Fµ(J)|u|C([T1,T2];C(Ω,RN))1+|v|Fµ(J)≤ε|v|Eu,µ(J)+Cε,
whereεisarbitrary.Ifwecombinetheaboveestimateswith(4.4.3)andchooseεsufficiently
small,thenwemaysubtractε|v|Eu,µ(J)onbothsidesoftheinequality,toobtain
|v|Eu,µ(J)|u|C([T,T];Cα(Ω,RN))1+|u(T1,u0)|Wqσ(Ω,RN).
21

162

AttractorsinStrongerNormsforRobinBoundaryConditions

Togetherwith(4.4.2)thisyieldstheassertedestimate.Inthesemilinearcasetheconstant
doesnotdependontheHöldernormofthesolution,sincethenonlytheterms|f(v)|E0,µ(J)
and|g(v)|Fµ(J)in(4.4.3)areestimated.
Weusetheaboveestimatetogiveasufficientconditionfortheexistenceofaglobal
attractorof(4.1.1)inthephasespaceMp2−2/pintermsoflowernorms.Thisisthemain
resultofthischapter.
Theorem4.4.2.Supposethat2the−2re/pareα,C>0suchthatforeachsolutionu(∙,u0)of
(4.1.1)withinitialvalueu0∈Mpitholds
t→limt+(supu0)|u(t,u0)|Cα(Ω,RN)≤C.
Then(4.1.1)hasaglobalattractorinMp2−2/p.
Proof.Wefirstshowthatt+(u0)=+∞forallu0∈Mp2−2/p.Assumethecontrary,
i.e.,t+(u0)<+∞.ThenLemma4.4.1andtheembeddingCα(Ω,RN)→Wpσ(Ω,RN)for
σ∈(0,α)yield
t∈[0,tsup+(u))|u(t,u0)|Wp2−2/p(Ω,RN)1+t∈[0,t+sup(u)/2)|u(t,u0)|Cα(Ω,RN),
00whichmeansthattheorbit{u(t,u0)}t∈[0,t+(u0))isboundedinWp2−2/p(Ω,RN).Itthus
hasaconvergentsubsequenceinWps(Ω,RN)fors∈(1+n/p,2−2/p),whichleadstoa
contradictiontothemaximalexistencetime,andthereforet+(u0)=+∞.Nowanother
applicationofLemma4.4.1yieldsthatthereisC0>0with
tlim→∞sup|u(t,u0)|Wp2−2/p(Ω,RN)≤C0
forallu0∈M2p−2/p.Thereforetheglobalsemiflowgeneratedby(4.1.1)hasanabsorbant
ballinMp2−2/p.SincethesemiflowisalsocompactbyTheorem4.3.6,theexistenceofa
globalattractorfollowsfrom[16,Corollary1.1.6].
Weconsiderspecialcasesof(4.1.1),whereanabsorbingsetinaweakernormissufficientfor
anattractorinMps.Westartwiththesemilinearcasewithnonlinearboundaryconditions.
Corollary4.4.3.Assumethat(aij)doesnotdependonu,andsupposethatthereare
q∈(1,∞),σ∈(0,22−−22/p/q]andaconstantC>0suchthatforeachsolutionu(∙,u0)of
(4.1.1)withu0∈Mpitholds
t→limt+(usup)|u(t,u0)|Wqσ(Ω,RN)∩L∞(Ω,RN)≤C.
0Then(4.1.1)hasaglobalattractorinMp2−2/p.
Proof.Lemma4.4.1yieldsaconstantC0suchthat
t→limt+(usup0)|u(t,u0)|Wq2−2/q(Ω,RN)≤C0(4.4.4)

(4.4.4)

4.4GlobalAttractorsinStrongerNorms163
forallu0∈Mp2−2/p.Weemployabootstrappingproceduretoshowthat(4.4.4)remains
trueifonereplacesWq2−2/q(Ω,RN)byCα(Ω,RN)withsomeα>0,andC0byapossibly
largerconstant.ItthenfollowsfromTheorem4.4.2that(4.1.1)hasaglobalattractorin
Mp2−2/pasasserted.Sobolev’sembeddingyields
Wq2−2/q(Ω,RN)→Cα(Ω,RN)
forsomeα>0ifq>n/2+1,andwearedoneinthiscase.Otherwise,incaseq∈
(1,n/2+1),weemploy
Wq2−2/q(Ω,RN)→Wqτ1(Ω,RN),
whichisvalidforsomesmallτ>0ifq1∈q,n+2nq−2q.Noteherethatn+2nq−2q>q
forallnandq∈(1,n/2+1).AnotherapplicationofLemma4.4.1yields(4.4.4)with
Wq2−2/q(Ω,RN)replacedbyWq21−2/q1(Ω,RN).Iteratively,thisyieldsastrictlyincreasing
sequenceofnumbersqkaslongasqk<n/2+1.Butsinceqk≥nn+2(1−−δ2)qkqforsmallδ>0
aslongasqk<n/2+1andn+2n−2q>1,thesequenceqkbecomeslargerthann/2+1
afterfinitelymanysteps.Thus(4.4.4)holdstruewithaHöldernorm,andthisfinishesthe
of.proForN=2wenextconsiderfortheunknownu=(u1,u2)quasilinearcross-diffusion
formtheofsystems∂tu1=divP(u)u1+R(u)u2+f1(u)inΩ,t>0,
∂tu2=divQ(u2)u2+f2(u)inΩ,t>0,
∂νu=0onΓ,t>0,(4.4.5)
u(0,∙)=u0inΩ.
Thisproblemfitsintooursettingwitha(u)=P(u)R(u),αij=δijandg=0.
0Q(u2)
WecanusetheresultsofKuiper&Dung[61]toweakenthenormfortheabsorbingball
considerably.Weassumethefollowingonthecoefficientsof(4.4.5).Therearenonnegative
continuousfunctionsΦ1,Φ2andconstantsC,d>0suchthatforallζ=(ζ1,ζ2)∈R2it
holdsP(ζ)≥d(1+ζ1),ζ1≥0,|R(ζ)|≤Φ1(ζ2)ζ1,Q(ζ2)≥d;
thepartialderivativesofP,Raremajorizedbysomepowersofζ1,ζ2;
|f(ζ)|≤Φ2(ζ2)(1+ζ1),g(ζ)ζ1r≤Φ2(ζ2)(1+ζ1r+1),forallζ1,ζ2≥0,r>0.
Corollary4.4.4.Undertheaboveassumptions,letthesolutionsof(4.4.5)benonnegative
fornonnegativeinitialdata.Supposethattherearer>n/2andC>0suchthatforall
u0∈Mp2−2/pitholds
tlim→∞sup|u1(t,u0)|Lr(Ω,R2)+|u2(t,u0)|L∞(Ω,R2)≤C.
Then(4.4.5)hasaglobalattractorinMp2−2/p.IfQdoesnotdependonu2onecantake
.1=r

164

AttractorsinStrongerNormsforRobinBoundaryConditions

Proof.ItisshownintheTheorems7and8of[61]that(4.4.5)hasaglobalattractorin
Wp1(Ω,R2)forallp∈(n+2,∞),fromwhichtheexistenceofanabsorbantsetinaCα-
normfollowsfromSobolev’sembedding.TheassertionisthusaconsequenceofTheorem
4.4.2.Incaseofasingleequation,N=1,thenormfortheabsorbantsetcanbeweakendupto
L1,usingestimatesofDeGiorgi-Nash-Mosertype.
Corollary4.4.5.Considerforu(t,x)∈Rtheproblem
∂tu=div(a(u)u)+f(u)inΩ,t>0,
∂νu=g(u)onΓ,t>0,(4.4.6)
u(0,∙)=u0inΩ,
wherea,fandgareassumedtobeboundedandthatthereexistsδ>0witha(ζ)≥δfor
allζ∈R.IfthereisaconstantC>0suchthatforeachu0∈Mp2−2/pthesolutionu(∙,u0)
satisfies(4.4.6)oft→limt+(usup)|u(t,u0)|L1(Ω)≤C,
0then(4.4.6)hasaglobalattractorinMp2−2/p.
Proof.Itisshownin[33,Theorem1]thattheexistenceofanabsorbantballinL1(Ω)
impliestheexistenceofanabsorbantballinL∞(Ω).Thisinturnyieldsanabsorbantball
inaHöldernorm,see[28,TheoremIII.1.3]or[34,Corollary4.2],andtheassertionfollows
4.4.2.Theoremfrom

ionsApplicat4.5Weapplytheresultsofthelastsectiontoshowconvergencetoattractorsinstrongernorms
dels.moconcretefor

4.5.1Reaction-DiffusionSystemswithNonlinearBoundaryConditions
Inaseriesofpapers,Carvalhoet.al.[15]consideredglobalattractorsforsemilinear
reaction-diffusionsystemswithnonlinearboundaryconditionsoftheform
∂tu−Δu=f(u)inΩ,t>0,
∂νu=g(u)onΓ,t>0,(4.5.1)
u(0,∙)=u0inΩ.
Herethesmoothnonlinearitiesf,g:RN→RNaredissipativeinthesensethatthereare
realnumberscianddiwith
limsupfi(ξ)<ci,limsupgi(ξ)<di,
|ξi|→∞ξi|ξi|→∞ξi

Applications4.5

165

suchthatthefirsteigenvalueλ0ofthelinearellipticproblem
−Δv−cv=λvinΩ,
∂νv−dv=0onΓ,
ispositive,wherec=(c1,...,cN)andd=(d1,...,dN).Thediscussionin[15,Section6]
showsthatthefirsteigenvalueoftheaboveproblemcanbepositivealthoughciordihasthe
‘wrong‘sign,i.e.,ispositive.Inthissensefcancompensateapossiblenondissipativeness
ofg,andviceversa.
In[15,Theorem4.1]itisshownthatundertheaboveassumptions(4.5.1)hasaglobal
attractorinthephasespaceW21(Ω,RN)∩C(Ω,RN).Corollary4.4.3improvesthisresult
ws.ollofasTheorem4.5.1.Undertheaboveassumptions,forp∈(n+2,∞)thesemiflowgenerated
by(4.5.1)hasaglobalattractorinthenonlinearphasespace
u0∈Wp2−2/p(Ω,RN):∂νu0=g(u0)onΓ.
4.5.2AChemotaxisModelwithVolume-FillingEffect
Foru(t,x),v(t,x)∈Rthefollowingchemotaxismodelwithvolume-fillingeffectwasintro-
ducedbyHillen&Painter[53],
∂tu=d1Δu−divuq(u)χ(v)v+uf(u)inΩ,t>0,
∂tv=d2Δv+g1(u)−vg2(v)inΩ,t>0,
∂νu=∂νv=0onΓ,t>0,
u(0,∙)=u0,v(0,∙)=v0inΩ,(4.5.2)
Thismodelmaybecastintheform(4.4.5)andisthusofseparateddivergenceform.Itis
assumedthatqisgivenby
q(u)=1−u/UM,UM>0,
andfurtherthatd1,d2>0forthediffusioncoefficientsand
f|(UM,∞)≤0,g1,g2≥0,g1(0)=0,vlim→∞vg2(v)→+∞,
forthesmoothreactiontermsf,g1andg2.Besidessmoothnessthereisnostructural
assumptionthesensitivityfunctionχ.Itmayevenchangeitssign.Wrozsek[87,88]showed
thatundertheseassumptions(4.5.2)possessesaglobalattractorinthephasespaces
(u0,v0)∈Wp1(Ω,R2):0≤u0≤UM,0≤v0,p∈(n,∞),
andfurtherthattheω-limitsetofeachsolutionorbitconsistsentirelyofequilibriawhich
satisfyacertainnonlocalproblem.Jiang&Zhang[56]showedthatinfacteverysolutionof
(4.5.2)convergestoanequilibrium.ItiswellknownthatforUM=∞blow-upofsolutions
mayoccuriftheinitialmassofu0istoolarge,cf.thesurveyarticle[54].ForUM<∞the

166

AttractorsinStrongerNormsforRobinBoundaryConditions

chemotacticterminthefirstequationbecomesbecomessmallifuisclosetoUM,which
preventssolutionsfromblow-up.
UsingLemma4.4.1,thesameargumentsasintheproofofTheorem4.4.2yieldthefollowing
improvementoftheresultof[87].
Theorem4.5.2.Undertheaboveassumptions,forp∈(n+2,∞)thechemotaxismodel
(4.5.2)hasaglobalattractorinthephasespace
(u0,v0)∈Wp2−2/p(Ω,R2):0≤u0≤UM,0≤v0.
4.5.3APopulationModelwithCross-Diffusion
OurlastexampleistheShigesada-Kawasaki-Teramotocross-diffusionmodelforpopulation
dynamics,introducedin[76],whichisforu(t,x),v(t,x)∈Rgivenby
∂tu=Δd1+α11u+α12v)u+u(a1−b1u−c1v)inΩ,t>0,
∂tv=Δd2+α21u+α22v)v+v(a2−b2u−c2v)inΩ,t>0,
∂νu=∂νv=0onΓ,t>0,
u(0,∙)=u0,v(0,∙)=v0inΩ.(4.5.3)
Againthismodelmaybecastintheform(4.4.5).Heretheconstantsai,bi,ci,di,i=1,2,
arepositive,andtheconstantsαij,i=1,2,arenonnegative.In[61,Theorem2]itisshown
that(4.5.3)hasaglobalattractorasadynamicalsysteminWp1(Ω,R2)forp∈(n,∞),
providedα22=0.Forn=2thisremainstruealsoforα22>0.Theorem4.4.2improves
ws.folloasthisTheorem4.5.3.Undertheaboveassumptions,for2−p2∈/p(n+22,∞)thepopulationmodel
(4.5.3)hasaglobalattractorinthephasespaceWp(Ω,R).

5Chapter

ofConditionsBoundaryReactive-Diffusive-ConvectiveType

nductiotroIn5.1Inthischapterweinvestigatelinearandquasilinearparabolicsystemswithdynamical
boundaryconditionsofreactive-diffusive-convectivetype.Fortheunknownu=u(t,x)∈
RN,whereN∈N,weconsidertheproblem1
∂tu=∂i(a1(u)∂iu)+a2(u)u+f(u)inΩ,t>0,
∂tu+b(∙,u)∂νu=divΓ(c1(∙,u)Γu)+c2(∙,u)Γu+g(∙,u)onΓ,t>0,
u(0,∙)=u0inΩ.(5.1.1)
ItisassumedthatΩ⊂RnisaboundeddomainwithsmoothboundaryΓ=∂Ω,where
n≥2.TheouternormalunitfieldandthenormalderivativeonΓaredenotedbyνand
∂ν=νtrΩ,respectively.ThespatialtraceonΩisdesignatedbytrΩ.Further,Γand
divΓarethesurfacegradientandthesurfacedivergenceonΓ,respectively.Weassume
thatthecoefficientsaresmooth,andthatforallx∈Γandζ∈RNitholds
a1(ζ),b(x,ζ)∈B(RN),c1(x,ζ)∈RN,
a2(ζ),c2(x,ζ)∈B(RN×n,RN),f(ζ),g(x,ζ)∈RN.
ThetermdivΓ(c1(∙,u)Γu)ismeantinwaythatitsk-thcomponentisgivenby
divΓ(c1(∙,u)Γu)k:=divΓ(c1k(∙,u)Γuk),k=1,...,N.
Thesystem(5.1.1)consistsoftwodynamicequation,coupledinapossiblynonlinearway
bythefluxtermb(∙,u)∂νu.ThetermdivΓ(c1(∙,u)Γu)takesintoaccountsurfacediffusion
effectsontheboundary,wherethetangentialfluxvectorJΓk=−c1k(∙,u)Γukofukmay
dependnonlinearlyonthesurfacegradientofuk.Forc1≡1oneobtainstheLaplace-
BeltramioperatorΔΓ=divΓΓonΓ.Further,thetermc2(∙,u)Γudescribesnonlinear
surfaceconvectionontheboundary.
1Weuseagainsumconvention.

168

BoundaryConditionsofReactive-Diffusive-ConvectiveType

Weexplainthedifferentialoperatorsontheboundary.Viathedirectionalderivativeat
x∈Γafunctionv∈C∞(Γ)inducesanelementofthedualspaceofTxΓ.Thesurface
gradientΓv(x)∈RNofvatxisthentheuniquecorrespondingelementofTxΓgiven
bytheRieszisomorphism,ifoneconsidersTxΓasaHilbertspacewiththescalarproduct
inducedfromRn.InlocalcoordinatesgforΓ,withfundamentalformG=(gij)and
inverseG−1=(gij),thecomponentsofthesurfacegradientwithrespecttothebasis
{∂1g,...,∂n−1g}ofthetangentialspacearegivenbythecomponentsofG−1n−1(v◦g)T,
i.e.,1−nΓv◦g=gij∂j(v◦g)∂ig.
=1i,jForatangentialvectorfieldw∈C∞(Γ,Rn),i.e.,w(x)∈TxΓforx∈Γ,thefunction
divΓw∈C∞(Γ)isincoordinatesggivenby
1−nGdivΓw◦g=1∂i|G|wi◦g,
||=1iwherewiarethecomponentsofwwithrespecttothebasis{∂1g,...,∂n−1g}.Forthe
componentsofthesurfacediffusiontermin(5.1.1)wethushave
n−1
divΓ(c1(∙,u)Γu)k◦g=1∂ic1k(∙,u)◦g|G|gij∂j(uk◦g),k=1,...,N.
|G|i,j=1
ThesearewelldefineddifferentialoperatorsonΓ(cf.AppendixA.5),withprincipalparts
equaltoc1kΔΓ,respectively.
Weimposethefollowingstructuralconditionsona1,bandc1,whereδ>0isindependent
ofx∈Γandζ∈RN.ByAkkwedenotethek-thdiagonalentryofamatrixA.
a1(ζ),b(x,ζ)areuppertriangularmatrices,c1k(x,ζ)≥δ,k=1,...,N;(5.1.2)
a1kk(ζ)≥δ,andeitherbkk(x,ζ)≥δorbkk(x,ζ)≤−δ,k=1,...,N.
Weemphasizethatthesignofthediagonalentriesofbmaychangefromlinetoline.
Letusdescribetheresultsandtheorganizationofthischapter.Forp∈(1,∞)welet
X0=Lp(Ω,RN)×Wp1−1/p(Γ,RN),
X1=(v,vΓ)∈Wp2(Ω,RN)×Wp3−1/p(Γ,RN):trΩv=vΓ,
andlookforsolutionsuinthemaximalregularityclass
Eu(J)=Wp1(J;X0)∩Lp(J;X1),
whereJ=(0,T)isafinitetimeinterval,T>0.Identifyingafunctionuwiththepair
(u,trΩu),herewewriteu∈Eu(J),withaslightabuseofnotation.InSection5.2wefirst
considerthelinearinhomogeneous,nonautonomousversionof(5.1.1)andshowthatit
enjoysthepropertyofmaximalLp,µ-regularityonfiniteintervals,verifyingtheconditions

5.2MaximalLp,µ-RegularityfortheLinearizedProblem

169

ofTheorem3.1.4.WethenturninSection5.3tothequasilinearcaseandshowthatfor
eachinitialvalueu0=∼(u0,trΩu0)fromthelinearphasespace
M=(v,vΓ)∈Wp2−2/p(Ω,RN)×Wp3−3/p(Γ,RN):trΩv=vΓ
thereisauniquemaximalsolutionu(∙,u0)∈C0,t+(u0);Mof(5.1.1),providedp∈
(n+2,∞).Heret+(u0)>0denotesthemaximalexistencetime.Weobtainstrongsolutions,
thatsensetheinu(∙,u0)∈Eu(0,τ)forallτ∈(0,t+(u0)).
Moreover,themapu0→u(∙,u0)definesacompactlocalsemiflowonM,whichhasthe
propertythatboundedorbitsarerelativelycompact.Theseresultsarebasedonmaximal
Lp-regularityforthelinearizationof(5.1.1),theregularitypropertiesofthesuperposition
operatorsoccurringin(5.1.1)andtherecentresultsof[59]onabstractquasilinearproblems
inLp,µ-spaces.Besidesthestructuralconditions(5.1.2)wedonothavetoimposeany
restrictionsonthenonlinearitiestoobtainthelocalsemiflow.Inparticular,wedonothave
toimposeanygrowthconditions.
WethenturntoglobalissuesandshowinSection5.4thatanaprioriHölderboundfora
solutionof(5.1.1)impliesthatitexistsgloballyintime.Weobtainthisresultbylocalizing
(5.1.1)inspaceandtime,employingagainthatthelinearizationof(5.1.1)hasmaximal
Lp-regularity,andbyperformingappropriateestimatesoftheresultingnonlinearerror
terms.InSection5.5wespecializetoasemilinearversionof(5.1.1),
∂tu=Δu+f(u)inΩ,t>0,
∂tu+∂νu=ΔΓu+g(u)onΓ,t>0,(5.1.3)
u(0,∙)=u0inΩ.
Underappropriatedissipativityconditionsonthereactiontermsfandgweobtaina
Lyapunovfunctionfor(5.1.3),thatalreadyappearedin[80],andaprioriestimatesinthe
energyspacesW21(Ω,RN)andW21(Γ,RN).ByaMoser-Alikakositerationprocedurewecan
showthatthisimpliesanaprioriL∞-bound,whichinturnleadstoglobalexistencefor
thesolutionsof(5.1.3).TheLyapunovfunction,togetherwithanotheraprioriestimate
fortheequilibriaof(5.1.3),yieldstheexistenceofaglobalattractorinM,andthateach
solutionconvergestothesetofequilibriaast→∞.
Problemsrelatedto(5.1.1)and(5.1.3)wereconsidered,forinstance,in[38,39,40,80,83].
Werefertotheintroductionofthisthesisformoreinformations.

5.2MaximalLp,µ-RegularityfortheLinearizedProblem
Inthissectionweshowthatthelinearizedversionof(5.1.1)enjoysmaximalLp,µ-regularity
byverifyingthenormalellipticitycondition(E)andtheLopatinskii-Shapirocondition(LS)
andusingTheorem3.1.4.Besidestheinterestinitsown,thislinearresultisthebasisfor
ourinvestigationofthequasilinearproblems.

170

BoundaryConditionsofReactive-Diffusive-ConvectiveType

Fortheunknownu=u(t,x)∈RNweconsiderlinearinhomogeneous,nonautonomous
parabolicsystemsoftheform
∂tu=A1Δu+A2u+A3u+finΩ,t∈J,
∂tu+B∂νu=C1ΔΓu+C2Γu+C3u+gonΓ,t∈J,(5.2.1)
u(0,∙)=u0inΩ,
trΩu(0,∙)=u0,ΓonΓ.
Asimplifiedversionof(5.2.1)wasconsideredinExample3.1.1.Herethecoefficientsmay
dependontandx,andareoftheform
A1(t,x),A3(t,x),B(t,x),C1(t,x),C3(t,x)∈B(RN),A2(t,x),C2(t,x)∈B(RN×n,RN).
Wefurtherimposethefollowingstructuralconditionsonthecoefficients,whicharesimilar
to(5.1.2).Thenumberδ>0isindependentoftandx.
A1(t,x),B(t,x),C1(t,x)areuppertriangularmatrices;fork=1,...,N:
A1kk(t,x),C1kk(t,x)≥δ,andeitherBkk(x,ζ)≥δorBkk(x,ζ)≤−δ.(5.2.2)
Wemaycast(5.2.1)intheform(3.1.1)bysetting
A(t,x,D)=−A1(t,x)Δ+A2(t,x)+A3(t,x),B0(t,x,D)=B(t,x)ν(x)trΩ,
C0(t,x,DΓ)=−C1(t,x)ΔΓ+C2(t,x)Γ+C3(t,x),B1=trΩ,C1=−1.
Forp∈(1,∞)thenontrivialpartoftheNewtonpolygonassociatedto(5.2.1),cf.Section
3.1,isthelinethroughtothepoints(0,3/2−1/2p)and(3−1/p,0).Thepoint(2,1/2−1/2p)
correspondingtotheoperatorC0liesonthisline,thepoint(0,1−1/2p)correspondingto
C1doesnot.
Toverify(E),notethattheprincipalsymbolofAisgivenby
A(t,x,ξ)=A1(t,x)|ξ|2,ξ∈Rn,(t,x)∈J×Ω.
SinceA1(t,x)isassumedtobepositivedefinite,thespectrumofA1(t,x)∈B(RN)is
containedintheright-halfplane.Hence(E)isvalid.

Problem(5.2.1)belongstoCase1,hencewedonothavetoconsiderasymptoticLopatinskii-
Shapiroconditions.Sincefurthertheunknownutakesvaluesinafinitedimensionalspace,
weonlyhavetoconsider(LS)withtrivialright-handsides.
Let(t,x)∈J×Γ,andtakecoordinatesgassociatedtox,cf.LemmaA.1.1.Thenthechart
(U,ϕ)correspondingtogsatisfiesϕ(x)=Oν(x),whereOν(x)isanorthogonalmatrixthat
rotatesν(x)to(0,...,0,−1)∈Rn.Forξ∈Rn−1andDy=−i∂ywethushave
Aνt,x,OνT(x)(ξ,Dy)=A1(t,x)(|ξ|2−∂y2),B0t,x,OνT(x)(ξ,∂y)=−B(t,x)∂y.

5.2MaximalLp,µ-RegularityfortheLinearizedProblem

171

Since(ϕ−1)(x)=OνT(x)itholdsG(x)=idn−1forthefundamentalformGcorrespondingto
g,i.e.,thecoordinatesgareorthonormalatx,andtheLaplace-Beltramioperatorreduces
toΔΓu(x)=Δn−1(u◦g)◦g−1(x).Thisyields
Cg0(t,x,ξ)=C1(t,x)|ξ|2,ξ∈Rn−1.
ByconventionwefurtherhaveC1=0,sincethepoint(0,1−1/2p)correspondingtoC1
doesnotlieonthenontrivialpartoftheNewtonpolygon.
Wethushavetoshowthatforallλ∈C+\{0}andξ∈Rn−1withλ+|ξ|=0theonly
solution(v,σ)oftheordinaryinitialvalueproblem
λ+A1(t,x)(|ξ|2−∂y2)v(y)=0,y>0,
−B(t,x)∂yv(0)+λ+C1(t,x)|ξ|2σ=0,(5.2.3)
,0=(0)vwherevisdecayingasy→∞isthetrivialone,i.e.,(v,σ)=(0,0).Solet(v,σ)solve
(5.2.3)andletvbedecaying.Wewritev=(v1,...,vN)andσ=(σ1,...,σN).Wenowmake
useofthetriangularstructureofA1,BandC1.DenotingbyA1NN(t,x)>0thediagonal
entryofA1(t,x)intheN-throw,weobtainthatvNsolves
λ/A1NN(t,x)+|ξ|2−∂y2vN=0,y>0,vN(0)=0,
whichimpliesvN≡0.Consequently−BNN(t,x)∂yvN(0)=0,whichshowsthatthesign
ofBNN(t,x)hasnoinfluenceonthevalidityof(LS).WethusobtainthatσNsatisfies
λ+C1NN(t,x)|ξ|2σN=0.SinceweassumeC1NN(t,x)>0andthatλandξdonot
vanishsimultaneously,itfollowsthatσN=0.Iteratingtheseargumentsandusingthe
diagonalstructureweobtainthateachcomponentofvandσvanishes.Hereagainthesign
ofthediagonalentriesofBhasnoinfluence.Thisverifies(LS)for(5.2.3).
Forthesolvabilityof(5.2.1)thecompatibilitycondition
g(0,∙)+B(0,∙)∂νu0,−C1(0,∙)ΔΓu0,Γ
−C2(0,∙)Γu0,Γ−C3(0,∙)u0,Γ∈B2(p,pµ−1/p)−1−1/p(Γ,RN)
m2(ustµ−1b/pe)−1sa−1/ptisfiedif2(µ−1/p)>1+1/p.SincethetracespaceofFµequals
Bp,p(Γ,RN)itsuffices,forinstance,if
B(0,∙),C1(0,∙),C2(0,∙),C3(0,∙)arepointwisemultipliersonB2(p,pµ−1/p)−1−1/p(Γ,RN).
(5.2.4)Notethatifincase2(µ−1/p)>1+n/p,whichisrelevantforthetreatmentofquasilinear
problems,thecoefficientssatisfythefirstconditionin(SB)and(SC),i.e.,
B,C1,C2,C3∈Fµ(J,F),
whereFstandsforB(RN)orB(RN×n,RN),then(5.2.4)isvalidbyLemma1.3.19and
eddings.bemolev’sSobTheaboveconsiderations,Theorem3.1.4and[26,Theorem2.2]yieldthefollowingresult.

172

BoundaryConditionsofReactive-Diffusive-ConvectiveType

Theorem5.2.1.LetJ=(0,T)beafiniteinterval,p∈(1,∞)andµ∈(1/p,1].Assume
thatthecoefficientsof(5.2.1)satisfy(5.2.2),(SD),(SB)and(SC),andfurther(5.2.4)if
2(µ−1/p)>1+1/p.Then(5.2.1)hasauniquesolutionusatisfying
u∈W1p,µ(J;Lp(Ω,RN))∩Lp,µ(J;Wp2(Ω,RN))
trΩu∈W3p,µ/2−1/2p(J;Lp(Γ,RN))∩Lp,µ(J;Wp3−1/p(Γ,RN)),
ifandonlyifthedataissubjectto
f∈Lp,µ(J;Lp(Ω,RN)),u0∈B2(p,pµ−1/p)(Ω,RN),u0,Γ∈B2(p,pµ−1/p)+1−1/p(Γ,RN),
g∈W1p,µ/2−1/2p(J;Lp(Γ,RN))∩Lp,µ(J;Wp1−1/p(Γ,RN)),
anditholdstrΩu0=u0,Γif2(µ−1/p)>1/p.Intheautonomouscase,i.e.,ifthecoefficients
donotdependont,therealizationoftheoperator
A1Δ+A2+A30
A=−B∂νC1ΔΓ+C2Γ+C3
onLp(Ω,RN)×Wp1−1/p(Γ,RN)withdomain
D(A)=(v,vΓ)∈Wp2(Ω,RN)×Wp3−1/p(Γ,RN):trΩv=vΓ
isthegeneratorofananalyticC0-semigroup.

Remark5.2.2.a)Thetheoremisthebasisforourinvestigationsofquasilinearproblems.
b)Itshouldbepossibletoverify(LS)for(5.2.1)undermoregeneralstructuralassumptions
ts.efficiencotheonc)Havingverified(E)and(LS),intheautonomouscasethemaximalLp,µ-regularity
resultalsofollowsfromtheresultintheunweightedcase[26,Theorem2.1]combinedwith
Theorem1.2.2ontheindependenceofmaximalLp,µ-regularityofµ∈(1/p,1].
d)Ifµissufficientlysmallsuchthatthespatialtraceofu0doesnotnecessarilyexistthere
mustbenorelationbetweentheinitialvaluesu0andu0,Γ.
e)ThesignofthediagonalentriesofBcanchangefromrowtorow.Thereasonisthat
B∂νisoflowerorderwithrespecttoC1ΔΓ.Thefactthatthesignhasnoinfluencecan
alsobeseenintheverificationoftheLopatinskii-ShapiroConditionabove.
f)Thetheoremgivespartialanswerstotheopenquestionsposedin[83].Inthispaperthe
problem

∂tu=ΔuinΩ,t>0,
∂tu−∂νu=ΔΓuonΓ,t>0,(5.2.5)
u(0,∙)=u0inΩ,
isstudied,anditisshownthat(5.2.5)generatesananalyticquasi-contractivesemigroup
spaceenergytheinH=(v,vΓ)∈W21(Ω)×W21(Γ):trΩv=vΓ.
Oursettingdiffersfromtheonein[83].Forp=2Theorem5.2.1yieldsasemigroupin
L2(Ω)×W21/2(Γ)for(5.2.5).

5.3TheLocalSemiflowforQuasilinearProblems

173

5.3TheLocalSemiflowforQuasilinearProblems
Thefunctionalanalyticalsettingforthesolutionsof(5.1.1)isasfollows.Forp∈(3/2,∞)
weconsidertheBanachspaces
X0:=Lp(Ω,RN)×Wp1−1/p(Γ,RN),E0(J):=Lp(J;X0),
X1:=(v,vΓ)∈Wp2(Ω,RN)×Wp3−1/p(Γ,RN):trΩv=vΓ,
Eu(J):=Wp1(J;X0)∩Lp(J;X1),
andwewritev=∼(v,vΓ)∈Eu(J).Recallfurtherthephasespace
M:=(v,vΓ)∈Wp2−2/p(Ω,RN)×Wp3−3/p(Γ,RN):trΩv=vΓ,
whichweconsiderasaclosedsubspaceofWp2−2/p(Ω,RN)×Wp3−3/p(Γ,RN).Denotingby
(∙,∙)s,ptherealinterpolationfunctor,s∈(0,1),wehavethefollowingcharacterizationof
.MLemma5.3.1.Forp∈(3/2,∞)itholdsthatM=(X0,X1)1−1/p,p.
Proof.DefineonX0theoperatorAbyAu=(−Δu,∂νu−ΔΓuΓ),whereu=(u,uΓ)∈
D(A):=X1.CombiningExample3.2andTheorem2.2of[26]weobtainA∈MRp(0,1)
forallp∈(1,∞),whichimpliesthattheCauchyProblem
∂tu+Au=0,t∈(0,1),u(0)=u0,
hasauniquesolutionu∈Eu(0,1)ifandonlyifu0∈(X0,X1)1−1/p,p.Itfurtherfollows
fromLemma1.3.5thatthetemporaltracemapsEu(0,1)continuouslyinto(X0,X1)1−1/p,p.
Ontheotherhanditisshownin[26,Corollary2.3]thattheaboveCauchyproblemhas
auniquesolutioninEu(0,1)ifandonlyifu0∈M,providedp∈(3/2,∞),andTheorem
1.3.6yieldsthatthetemporaltracemaps
Eu(0,1)→Wp1(0,1;X0)∩Lp(0,1;Wp2(Ω,RN)×Wp3−1/p(Γ,RN))
continuouslyintoWp2−2/p(Ω,RN)×Wp3−3/p(Γ,RN).ThereforeMand(X0,X1)1−1/p,pco-
incideassets,andthemaximalregularityestimatesimpliedbyTheorem2.2,Corollary2.3
of[26]andthecontinuityofthetracesyield
|u0|(X0,X1)1−1/p,p|u|Eu(0,1)|u0|Wp2−2/p(Ω,RN)+|u0|Wp3−3/p(Γ,RN),
ersa.vviceandWedefinethemapsA:M→B(X1,X0)andF:M→X0by
−∂i(a1(u)∂iv)0
A(u)v:=b(∙,uΓ)∂νv−divΓ(c1(∙,uΓ)ΓvΓ),u∈M,v∈X1,
a2(u)u+f(u)
F(u):=c2(∙,uΓ)ΓuΓ+g(∙,uΓ),u∈M.

174

BoundaryConditionsofReactive-Diffusive-ConvectiveType

Asanabstractquasilinearevolutionequation,thesystem(5.1.1)takestheform
∂tu(t)+A(u(t))u(t)=F(u(t)),t>0,u(0)=u0.
WeshowthatthemapsAandFarelocallyLipschitzcontinuousforsufficientlylargep.
RecalltheSobolevembeddings
Wp2−2/p(Ω,RN)→C1(Ω,RN),Wp3−3/p(Γ,RN)→C2(Γ,RN),p>n+2.(5.3.1)
Lemma5.3.2.Forp∈(n+2,∞)thefunctions
A:M→B(X1,X0),F:M→X0,
areLipschitzcontinuousonboundedsubsetsofM.Moreover,underthestructuralcondi-
tions(5.1.2)wehavethatforallu0∈MandallfiniteintervalsJ=(0,T)theoperator
A(u0)onX0withdomainX1enjoysmaximalLp-regularityonJ.
Proof.(I)Theembeddings(5.3.1)showthatA(u)∈B(X1,X0)andF(u)∈X0for
u∈M.FortheregularityofA,weestimateforu,v∈Mandw∈X1with|w|X1≤1
|(A(u)−A(v))w|X0
n|(a1(u)−a1(v))∂iw|Wp1(Ω,RN)+|(b(∙,uΓ)−b(∙,vΓ))∂νw|Wp1−1/p(Γ,RN)
=1i+|(c1(∙,uΓ)−c1(∙,vΓ))ΓwΓ|Wp2−1/p(Γ,Rn)
|a1(u)−a1(v)|C1(Ω,RN×N)
+|b(∙,uΓ)−b(∙,vΓ)|C1(Γ,RN×N)+|c1(∙,uΓ)−c1(∙,vΓ)|C2(Γ,RN).
Itisnothardtoshowthatthesuperpositionoperators
u→a1(u),uΓ→b(∙,uΓ),uΓ→c1(∙,uΓ),
areLipschitzcontinuousonboundedsetsasmaps
C1(Ω,RN)→C1(Ω,RN×N),C1(Γ,RN)→C1(Γ,RN×N),C2(Γ,RN)→C2(Γ,RN),
respectively.Now(5.3.1)yieldsthatAisLipschitzcontinuousonboundedsubsetsofM.
SimilarargumentsshowtheassertedregularityofF.
(II)Letu0∈Mbegiven.Theembeddings(5.3.1)yieldthatforu0∈Mthecoefficientsof
A(u0)arecontinuousonΩandcontinuouslydifferentiableonΓ,respectively.Thus(SD),
(SB),(SC)and(5.2.4)arevalid.Theconditions(5.1.2)andTheorem5.2.1thusyieldthat
therealizationofA(u0)onX0withdomainX1enjoysmaximalLp-regularityonfinitetime
als.tervinAfterthesepreparationsweobtainlocalwell-posednessfor(5.1.1)fromtheresultsin[59].

5.4APrioriHölderBoundsimplyGlobalExistence

175

Theorem5.3.3.Assumethatthecoefficientsof(5.1.1)aresmooth,andthattheysatisfy
thestructuralconditions(5.1.2).Letfurtherp∈(n+2,∞).Thenforallinitialvalues
u0∈Mtheproblem(5.1.1)hasauniquemaximalsolution
u(∙,u0)∈C[0,t+(u0));M,
suchthatu(∙,u0)∈E1(0,τ)forallτ∈(0,t+(u0)),wheret+(u0)>0denotesthe
maximalexistencetime.Thesolutionmapu0→u(∙,u0)isalocalsemiflow2onM.If
{u(∙,u0)}t∈[0,t+(u0))isboundedinM,thent+(u0)=+∞andthecorrespondingorbitis
relativelycompactinM.
Proof.DuetoLemma5.3.2wemayapplytheTheorems2.1,3.1andRemark2.3of[59]
problemquasilinearabstracttheto∂tu(t)+A(u(t))u(t)=F(u(t)),t>0,u(0)=u0∈M,
whichisequivalentto(5.1.1).Henceallassertionsfollow,exceptthecompactnessproperty
ofthesolutionmap.Using[59,Theorem2.1],theproofofthisfactiscompletelyanalogous
totheproofofProposition4.3.5.

5.4APrioriHölderBoundsimplyGlobalExistence
WeshowhowmaximalLp-regularitycanbeusedtoreducethequestionofglobalexistence
ofsolutionstotheboundednessinaHöldernorm.

Theorem5.4.1.UndertheassumptionsofTheorem5.3.3,letu(∙,u0)bethemaximalsolu-
tionof(5.1.1)withinitialvalueu0∈M.Ifu(∙,u0)isuniformlycontinuousin[0,t+(u0))×Ω
holdsitandt∈[0,t+sup(u0))|u(t,u0)|Cα(Ω,RN)<+∞
forsomeα>0,thenu(∙,u0)existsglobally,t+(u0)=+∞,providedpissufficientlylarge.
Proof.Theplanistoshowthatift+(u0)<+∞thentheorbitisboundedinM,provided
pissufficientlylarge.ThisleadstoacontradictiontoTheorem5.3.3,andshowsthatt+(u0)
cannotbefinite.Assumethatt+(u0)<+∞,anddenotebyu=u(∙,u0)thesolutionof
(5.1.1).Wewillshowthatforsufficientlysmallη>0thequantity|u|E1(tη,T),where
tη:=t+(u0)−η,
isboundedbyaconstantindependentofT∈(tη,t+(u0)).ThensupT∈[0,t+(u0))|u(T)|Mis
finite,andwearedone.
(I)Welocalizetheprobleminspace.Duetoitsuniformcontinuityon[0,t+(u0))wemay
continueu=u(∙,u0)toaboundeduniformlycontinuousfunctionon[0,t+(u0)]×Ω.Thus
forgivenε>0thereareη,δ>0with
|u(t,x)−u(s,y)|<εfor|x−y|<δ,|t−s|<η,x,y∈Ω,t,s∈[0,t+(u0)].(5.4.1)
2WerefertoSection4.3foraprecisedefinitionofalocalsemiflow.

176BoundaryConditionsofReactive-Diffusive-ConvectiveType
Forδ>0wechooseafinitenumberofpointsxl∈ΩsuchthatlBδ(xl)coversΩ,anda
partitionofunity{ψl}forΩsubordinatetothiscover.Givenδ>0weobtain
|u|E1(tη,T)≤|ψlu|E1(tη,T).(5.4.2)
lThefunctionusolvesthelinearnonautonomousproblem3
∂tv−∂ia1(u)∂iv=a2(u)u+f(u)=:f1inΩ×(tη,T),
∂tv+b(∙,u)∂νv−divΓ(c1(∙,u)Γv)=c2(∙,u)Γu+g(∙,u)=:g1onΓ×(tη,T),
v(tη,∙)=u(tη,∙)inΩ.
Thusforeachlthefunctionw:=ψlusatisfies
∂tw−∂ia1(u)∂iw=ψlf1−[∂i(a1(u)∂i),ψl]u
=:ψlf1+f2inΩ×(tη,T),
∂tw+b(∙,u)∂νw−divΓ(c1(∙,u)Γw)=ψlg1+b(∙,u)[∂ν,ψl]u−[divΓ(c1(∙,u)Γ),ψl]u
=:ψlg1+g2onΓ×(tη,T),
w(tη,∙)=ψlu(tη,∙)inΩ.
Localizinginspaceandtime,weobtainthatwsatisfies
∂tw−a1(u(tη,xl))Δw=ψlf1+f2+∂ia1(u)−a1(u(tη,xl))∂iw
=:ψlf1+f2+f3inΩ×(tη,T),
∂tw+b(xl,u(tη,xl))∂νw−c1(xl,u(tη,xl))ΔΓw
=ψlg1+g2+b(∙,u)−b(xl,u(tη,xl))∂νw
+divΓ(c1(∙,u)−c1(xlu(tη,xl)))Γw
=:ψlg1+g2+g3onΓ×(tη,T),
w(tη,∙)=ψlu(tη,∙)inΩ.
BythemaximalregularityTheorem1.2.3thereisaconstantC,whichdoesnotdepend
onT,ηandδ,suchthat
|w|E1(tη,T)≤C|ψlf1+f2+f3,ψlg1+g2+g3|E0(tη,T)+|ψlu(tη,∙)|M.(5.4.3)
AcompactnessargumentfurtheryieldsthatCisuniformin|u|BC([0,t+(u0)]×Ω,RN).Our
objectiveisnowtoshowthatforgivenσ>0anestimateoftheform
|ψlf1+f2+f3,ψlg1+g2+g3|E0(tη,T)≤σ|u|E1(tη,T)(5.4.4)
+C|u(∙,u0)|BC([0,t+(u0);Cα(Ω,RN)),δ,η,σ
isvalid.Ifwethencombine(5.4.3)with(5.4.2)andchooseσsufficientlysmallwemay
1subtract2|u|E1(tη,T)onbothsidesof(5.4.2)toobtaintheboundednessof|u|E1(tη,T)inde-
pendentofT.Throughoutwewrite|∙|∞foranyoccurringsup-norm.
3ThroughoutweneglectthesubscriptΓifuisconsideredontheboundary.

5.4APrioriHölderBoundsimplyGlobalExistence

177

(II)Thefunctionsψlf1,f2andf3mustbeestimatedintheLp(tη,T;Lp(Ω,RN))-norm.
Westartwiththeterm
ψlf1=ψla2(u)u+f(u).
Forthefirstsummandwehaveforgivenσ>0,usingtheinterpolationinequalityand
Young’sinequality,
|ψla2(u)u|Lp(tη,T;Lp(Ω,RN))≤C(|u|∞)|u|Lp(tη,T;Lp(Ω,RN×n))≤σ|u|E1(tη,T)+C(|u|∞,σ).
Thenexttermiseasilyestimatedby
|ψlf(u)|Lp(tη,T;Lp(Ω,RN))≤C(|u|∞).
Wenowconsiderthecommutatortermf2=[∂i(a1(u)∂i),ψl]u.Foreachiitholds
[∂i(a1(u)∂i),ψl]u=[a1(u)∂iu∂i,ψl]u+[a1(u)∂i∂i,ψl]u.
Asabovewehavethat
|[a1(u)∂iu∂i,ψl]u|Lp(tη,T;Lp(Ω,RN))=|∂iψla1(u)u∂iu|Lp(tη,T;Lp(Ω,RN))
≤C(|u|∞,δ)|u|Lp(tη,T;Wp1(Ω,RN))
≤σ|u|E1(tη,T)+C(|u|∞,δ,σ),

furtherand|[a1(u)∂i∂i,ψl]u|Lp(tη,T;Lp(Ω,RN))≤C(|u|∞,δ)|u|Lp(tη,T;Wp1(Ω,RN))
≤σ|u|E1(tη,T)+C(|u|∞,δ,σ).
Wenextconsiderthetermf3=∂ia1(u)−a1(u(tη,xl))∂i(ψlu).Foreachiwehave
∂ia1(u)−a1(u(tη,xl))∂i(ψlu)=a1(u)−a1(u(tη,xl))∂i∂i(ψlu)
+∂iψla1(u)u∂iu+ψla1(u)∂iu∂iu.
Forthefirstsummandweuse(5.4.1)toobtain
|a1(u)−a1(u(tη,xl))∂i∂i(ψlu)|Lp(tη,T;Lp(Ω,RN))
≤|a1(u)−a1(u(tη,xl))∂i∂iu|Lp(tη,T;Lp(Ω∩Bδ(xl),RN))
+C(|u|∞,δ)|u|Lp(tη,T;Wp1(Ω,RN))
≤σ|u|E1(tη,T)+C(|u|∞,δ,σ),
providedδandηaresufficientlysmall.Wefurtherhave,asbefore,
|∂iψla1(u)u∂iu|Lp(tη,T;Lp(Ω,RN))≤σ|u|E1(tη,T)+C(|u|∞,δ,σ).
Forthenexttermweobservethat,byHölder’sinequality,
Tp2p|ψla1(u)∂iu∂iu|Lp(tη,T;Lp(Ω,RN))≤C(|u|∞)|u(t)|W21p(Ω,RN)dt.

178

BoundaryConditionsofReactive-Diffusive-ConvectiveType

TheGagliardo-Nirenberginequality(PropositionA.6.2)yieldsanumbers∈(0,2),close
to2,suchthat
+ppp2|u(t)|W21p(Ω,RN)≤C|u(t)|Wps(Ω,RN)|u(t)|Cα(Ω,RN),t∈(tη,t(u0)).
Therefore,againbytheinterpolationinequalityandYoung’sinequality,
|ψla1(u)∂iu∂iu|Lp(tη,T;Lp(Ω,RN))≤C(|u|BC([0,t+(u0));Cα(Ω,RN)))|u|Lp(tη,T;Wps(Ω,RN))
≤σ|u|E1(tη,T)+C(|u|BC([0,t+(u0));Cα(Ω,RN)),σ).
Wehavethusestimatedthetermsψlf1,f2andf3asdesiredfor(5.4.4).
(III)Wenexttreatthetermsψlg1,g2andg3intheLp(tη,T;Wp1−1/p(Γ,RN))-norm.It
followsfromLemma1.3.20that
|ϕφ|Wp1−1/p(Γ,RN)|ϕ|∞|φ|Wp1−1/p(Γ,RN)+|ϕ|Wp1−1/p(Γ,RN)|φ|∞(5.4.5)
forallϕ∈Wp1−1/p(Γ,B(RN))andφ∈Wp1−1/p(Γ,RN).Itcanfurtherbeshownasinthe
proofofLemma4.2.3thatforasmoothfunctionh:Γ×RN→RNitholds
|h(∙,φ)|Wp1−1/p(Γ,RN)≤C(|φ|∞)1+|φ|Wp1−1/p(Γ,RN).(5.4.6)
Westartwith
ψlg1=ψlc2(∙,u)Γu+g(∙,u).
Forthefirstsummandwehave,using(5.4.5)and(5.4.6),
|ψlc2(∙,u)Γu|Lp(tη,T;Wp1−1/p(Γ,RN))≤C(δ)|c2(∙,u)Γu|Lp(tη,T;Wp1−1/p(Γ,RN×n))
≤C(|u|∞,δ)|u|Lp(tη,T;Wp2−1/p(Γ,RN))+C(δ)|c2(∙,u)|Lp(tη,T;Wp1−1/p(Γ,RN))|Γu|∞
≤σ|u|E1(tη,T)+C(|u|∞,δ,σ)+C(|u|∞,δ)(1+|u|Lp(tη,T;Wp1−1/p(Γ,RN)))|Γu|∞.
Forsufficientlysmallτ>0theGagliardo-Nirenberginequalityyields
θ−1θ|u(t)|Wp1+(n−1)/p+τ(Γ,RN)|u(t)|W3−3/p(Γ,RN)|u(t)|Cα(Ω,RN),
pprovidedθ∈(0,1)satisfies1<θ3−np+2+(1−θ)α.Thisinequalitycanbefulfilledby
11someθ<3−1/p,providedpissufficientlylargecomparedtoα.Weusethisfacttogether
withtheembeddingE1(tη,T)→L∞(tη,T;M)toobtain
|Γu|∞≤C|u|L∞(tη,T;Wp1+(n−1)/p+τ(Γ,RN))
θ≤C(|u|BC([0,t+(u0));Cα(Ω,RN)))|u|L∞(tη,T;Wp3−3/p(Γ,RN))
θ≤C(|u|BC([0,t+(u0));Cα(Ω,RN)),η)|u|E1(tη,T)(5.4.7)
1forsomeθ<3−1/p.ByYoung’sinequalitywethushave
C(|u|∞,δ)|Γu|∞≤σ|u|E1(tη,T)+C(|u|BC([0,t+(u0));Cα(Ω,RN)),δ,η,σ).

5.4APrioriHölderBoundsimplyGlobalExistence

Theinterpolationinequalityimpliesthat
13−−11/p/p
|u|Lp(tη,T;Wp1−1/p(Γ,RN))≤C(|u|∞)|u|E1(tη,T).
Combiningthisestimatewith(5.4.7),weobtain

179

(5.4.8)

C(|u|∞,δ)|u|Lp(tη,T;Wp1−1/p(Γ,RN))|Γu|∞
≤σ|u|E1(tη,T)+C(|u|BC([0,t+(u0));Cα(Ω,RN)),δ,η,σ),
whichfinishestheestimatesforthefirstsummandofψlg1.Using(5.4.5)and(5.4.6),the
secondsummandofψlg1isestimatedby
|ψlg(∙,u)|Lp(tη,T;Wp1−1/p(Γ,RN))≤C(|u|∞,δ)1+|u|Lp(tη,T;Wp1−1/p(Γ,RN))
≤σ|u|E1(tη,T)+C(|u|∞,δ,σ).
Wecontinuewiththecommutatorterm

g2=b(∙,u)[∂ν,ψl]u−[divΓ(c1(∙,u)Γ),ψl]u.
Forthefirstsummandweuseagain(5.4.5)and(5.4.6)toobtain

|b(∙,u)[∂ν,ψl]u|Lp(tη,T;Wp1−1/p(Γ,RN))≤C(δ)|b(∙,u)u|Lp(tη,T;Wp1−1/p(Γ,RN))
≤C(|u|∞,δ)1+|u|Lp(tη,T;Wp1−1/p(Γ,RN))
≤σ|u|E1(tη,T)+C(|u|∞,δ,σ).

Forthesecondsummandofg2wehave
|[divΓ(c1(∙,u)Γ),ψl]u|Lp(tη,T;Wp1−1/p(Γ,RN))(5.4.9)
|(ΔΓψl)c1(∙,u)u|Lp(tη,T;Wp1−1/p(Γ,RN))+|c1(∙,u)ΓψlΓu|Lp(tη,T;Wp1−1/p(Γ,RN))
+|Γc1(∙,u)+∂uc1(∙,u)Γu(Γ(ψlu)+ψlΓu)|Lp(tη,T;Wp1−1/p(Γ,RN)).
Herethefirstandthesecondsummandmaybetreatedasabove.Forthethirdsummand
weconcentrateontheterminvolvingΓ(ψlu).Weestimate
|Γc1(∙,u)Γ(ψlu)|Lp(tη,T;Wp1−1/p(Γ,RN))
≤C(|u|∞)|Γ(ψlu)|Lp(tη,T;Wp1−1/p(Γ,RN×n))
+C(1+|u|Lp(tη,T;Wp1−1/p(Γ,RN)))|Γ(ψlu)|∞
≤C(|u|∞,δ)|u|Lp(tη,T;Wp2−1/p(Γ,RN))
+C(δ)(1+|u|Lp(tη,T;Wp1−1/p(Γ,RN)))|u|L∞(tη,T;W∞1(Γ,RN)).

180

BoundaryConditionsofReactive-Diffusive-ConvectiveType

Notethatthefirstsummandisoflowerorder.Using(5.4.7)and(5.4.8)weobtainthat
alsothesecondtermisoflowerorder.Wefurtherhave
|∂uc1(∙,u)ΓuΓ(ψlu)|Lp(tη,T;Wp1−1/p(Γ,RN))
≤C(|u|∞)(1+|u|Lp(tη,T;Wp1−1/p(Γ,RN)))|ΓuΓ(ψlu)|∞
+C(|u|∞)|ΓuΓ(ψlu)|Lp(tη,T;Wp1−1/p(Γ,RN×n))
2≤C(|u|∞,δ)(1+|u|Lp(tη,T;Wp1−1/p(Γ,RN)))|u|L∞(tη,T;W∞1(Γ,RN))
+C(|u|∞,δ)|u|L∞(tη,T;W∞1(Γ,RN))|u|Lp(tη,T;Wp2−1/p(Γ,RN)).
Sinceθ<3−11/pitfollowsfrom(5.4.7),(5.4.8)andYoung’sinequalitythatherethefirst
termmaybeestimatedasdesired.Moreover,theinterpolationinequalityyields
32−−11/p/p
|u|Lp(tη,T;Wp2−1/p(Γ,RN))≤C(|u|∞)|u|E1(tη,T).
andthisfinishestheestimatesforthisterm.Wefinallyconsider
Combiningthiswith(5.4.7)weobtainthedesiredestimatealsoforthesecondtermofg2,
g3=b(∙,u)−b(xl,u(tη,xl))∂ν(ψlu)+divΓ(c1(∙,u)−c1(xl,u(tη,xl)))Γ(ψlu).
Forthefirstsummand,choosingδandηsufficientlysmallandusing(5.4.1),(5.4.5),(5.4.7)
and(5.4.8)weobtain
|b(∙,u)−b(xl,u(tη,xl))∂ν(ψlu)|Lp(tη,T;Wp1−1/p(Γ,RN))
≤C|ψl(b(∙,u)−b(xl,u(tη,xl)))|∞|∂νu|Lp(tη,T;Wp1−1/p(Γ,RN))
+C(|u|∞,δ)(1+|u|Lp(tη,T;Wp1−1/p(Γ,RN)))|u|L∞(tη,T;W∞1(Γ,RN))
≤σ|u|E1(tη,T)+C(|u|BC([0,t+(u0));Cα(Ω,RN)),δ,σ).
Wefurtherhave
|divΓ(c1(∙,u)−c1(xl,u(tη,xl)))Γ(ψlu)|Lp(tη,T;Wp1−1/p(Γ,RN))
≤|(c1(∙,u)−c1(xl,u(tη,xl)))ΔΓ(ψlu)|Lp(tη,T;Wp1−1/p(Γ,RN))
+|(Γc1(∙,u)+∂uc1(∙,u)Γu)Γ(ψlu)|Lp(tη,T;Wp1−1/p(Γ,RN)).
Choosingδandηsufficientlysmall,using(5.4.1),(5.4.5)andthetreatinglowerorderterms
asbefore,herethefirsttermmaybeestimatedasdesired.Thesecondtermisofthesame
typeistheonein(5.4.9).Thisfinishestheproof.
Itisnownaturaltoaskforsufficientconditionsonthenonlinearitiesin(5.1.1)thatguar-
anteeanaprioriHölderboundofsolutions.Thiswillbesubjecttofuturework.

5.5TheGlobalAttractorforSemilinearDissipativeSystems
fixwnoeW

p∈(n+2,∞)

5.5TheGlobalAttractorforSemilinearDissipativeSystems

181

andinvestigatethelong-timebehaviourofthefollowingsemilinearversionof(5.1.1),
∂tu=Δu+f(u)onΩ,t>0,
∂tu+∂νu=ΔΓu+g(u)onΓ,t>0,(5.5.1)
u(0,∙)=u0onΩ,
wherethereactiontermsf,g:RN→RNareassumedtobesmooth.Theorem5.3.3implies
that(5.5.1)generatesacompactlocalsemiflowofsolutionsinthephasespace
M=(v,vΓ)∈Wp2−2/p(Ω,RN)×Wp3−3/p(Γ,RN):trΩv=vΓ,
However,dueto[26,Theorem2.2],therealizationoftheoperator
0Δ−A=∂ν−ΔΓ
onX0=Lp(Ω,RN)×Wp1−1/p(Γ,RN)withdomain
D(A)=X1=(v,vΓ)∈Wp2(Ω,RN)×Wp3−1/p(Γ,RN):trΩv=vΓ
enjoysmaximalLp-regularityoneachfiniteintervalJ=(0,T),and−Ageneratesanana-
lyticC0-semigrouponX0.Thuslocalwell-posednessof(5.5.1)alsofollowsfromsemilinear
theory[51,Chapter3].Itisnowasimpleconsequenceofthevariationofconstantsformula
thatforu0∈Mthecorrespondingmaximalsolutionu(∙,u0)of(5.5.1)hastheadditional
ertiesoppryregularitu(∙,u0)∈C0,t+(u0);X1∩C10,t+(u0);M,(5.5.2)
see[16,Corollary2.3.1].Interestedinthelong-timebehaviourofsolutions,wemaythus
assumethatu0∈X1fortheinitialvaluesof(5.5.1).
Forq∈(1,∞)ands>0withs=1/qweintroducetheBanachspaces
sBqs,q(Ω,RN)×Bqs,q+1−1/q(Γ,RN),s<1/q,
Mq:=(u,uΓ)∈Bqs,q(Ω,RN)×Bqs,q+1−1/q(Γ,RN):trΩu=uΓ,s>1/q,
equippedwiththenormofBqs,q(Ω,RN)×Bqs,q+1−1/q(Γ,RN),respectively.Observethat
M=Mp2−2/p.
UsingthemaximalLp,µ-regularityTheorem5.2.1forA,onecanargueinthesamewayas
intheproofofLemma4.4.1toobtainthefollowingresult.Itshowsthatthesolutionina
strongnormcanbecontrolledbythesolutioninweakernorm.
Lemma5.5.1.Letq∈(1,p]andµ∈(1/q,1],sets=2(µ−1/q)∈(0,2−2/q],andassume
thats∈/{1/q,1+1/q}.Thenforτ>0thereisaconstantC=C(|u(∙,u0)|C([T1,T2]×Ω,RN),τ)
thathsuc|u(T2,u0)|M2q−2/q≤C1+|u(T1,u0)|Mqs
isvalidforall0<T1<T2<+∞withτ=T2−T1andallu0∈X1witht+(u0)<T2,
whereu(∙,u0)denotesthecorrespondingsolutionof(5.5.1).

182

BoundaryConditionsofReactive-Diffusive-ConvectiveType

Asaconsequencewehavethefollowingsufficientconditionsforglobalexistenceandrela-
orbits.compactelytivProposition5.5.2.Letu0∈X1,andsupposethatthecorrespondingsolutionof(5.5.1)
satisfiesu(∙,u0)∈BC([0,t+(u0))×Ω,RN).Thent+(u0)=+∞.Ifitadditionallyholds
that{u(t,u0)}t∈[0,∞)isboundedinMpsforsomes>0,then{u(t,u0)}t∈[0,∞)isrelatively
.incompactMProof.(I)Supposethatt+(u0)<∞.ItthenfollowsfromLemma5.5.1that
|u(T,u0)|Mt∈[0,t+sup(u0)/2)1+|u(t,u0)|M
forallT∈(t+(u0)/2,t+(u0)).ThustheorbitisboundedinM,whichcontradictsTheorem
5.3.3andyieldst+(u0)=+∞.
(II)Nowsupposeinadditionthat{u(t,u0)}t∈[0,∞)isboundedinMpsforsomes>0.
ThenanotherapplicationofLemma5.5.1yields
|u(T+1,u0)|M1+|u(T,u0)|Mps
forallT≥1.Thus{u(t,u0)}t∈[1,∞)isboundedinM,andtherelativecompactnessofthe
orbitfollowsagainfromTheorem5.3.3.
WenextwanttoestablishanL∞aprioriestimatefor(5.5.1)foraclassofreactionterms
f,g:RN→RN.Wefirstshowthatif
ζf(ζ)≤C(1+|ζ|2),ζg(ζ)≤C(1+|ζ|2),ζ∈RN,(5.5.3)
isvalidforaconstantC>0,thentheL∞-normofasolutioncanbecontrolledbyits
L1-norm.Observethat(5.5.3)isasignconditionforlarge|ζ|.WefurtherderiveanL∞-
estimatefortheequilibriaof(5.5.1)undertheaboveassumption.Notethatdueto(5.5.2),
eachequilibriumof(5.5.1)mustbelongtoX1→Wp2(Ω,RN)×Wp3−1/p(Γ,RN).
Lemma5.5.3.Assumethat(5.5.3)holdstrue.Thenforeachu0∈X1thereisaconstant
C1suchthatthecorrespondingsolutionu(∙,u0)of(5.5.1)satisfies
|u(∙,u0)|BC([0,t+(u0))×Ω,RN)≤C1max|u(∙,u0)|BC([0,t+(u0)),L1(Ω,RN)×L1(Γ,RN)),1.
Moreover,thereisaconstantC2>0suchthatforeachequilibriumu0∈X1of(5.5.1)it
holds|u0|BC(Ω,RN)≤C2max|u0|L1(Ω,RN)×L1(Γ,RN),1.
Proof.WeuseaMoser-Alikakositerationprocedure,presentedin[16,Section9.3]for
scalarproblemswithstaticboundaryconditions.Givent∈(0,t+(u0))andk∈N,the
planistofindanupperboundfortheL2k(Ω,RN)×L2k(Γ,RN)-normofu(t,u0),whichis
independentoftandk.Withaslightabuseofnotationwewriteu=u(t,u0)forfixedt.
Recallthatitholdsu∈X1by(5.5.2).

5.5TheGlobalAttractorforSemilinearDissipativeSystems

183

(I)WetakethescalarproductinRNofthedomainequation∂tu=Δu+f(u)attimet
with|u|2k−2u,integrateoverΩ,andintegratebyparts,toobtain
1dkkk
k|u|2dx=Δu∙|u|2−2udx+f(u)∙|u|2−2udx
2dtΩΩΩ
N=−ui∙(|u|2k−2ui)dx+∂νu∙|u|2k−2udσ(x)+f(u)∙|u|2k−2udx.
ΩΓΩ=1iThismanipulationisjustifiedduetou∈X1.Nowsupposethatk≥2.Fortheintegrand
ofthefirstsummandwehave
ui∙(|u|2k−2ui)=∂jui∂j(|u|2k−2ui)
NNn
i=1i=1j=1
nn=(2k−2)|u|2k−4|∂ju∙u|2+|u|2k−2|∂ju|2
=1j=1jnn≥(2k−2)|u|2k−4|∂ju∙u|2+|u|2k−4|∂ju|2|u|2(cos(∂ju,u))2
=1j=1jn=(2k−1)|u|2k−4|∂ju∙u|2.
=1jOntheotherhanditholds
(∂j|u|2k−1)2=|2k−1|u|2k−1−2∂ju∙u|2=22k−2|u|2k−4|∂ju∙u|2,
sothatweobtain
d|u|2kdx≤−(2k−1)22−k||u|2k−1|2dx
dtΩΩ
+2k∂νu∙|u|2k−2udσ(x)+2kf(u)∙|u|2k−2udx.
ΩΓNotethatthisestimateisalsotruefork=1,with||u|2k−1|2replacedby|u|2.Similarly,
takingthescalarproductinRNoftheboundaryequation∂tu=ΔΓu−∂νu+g(u)attime
twith|u|2k−2uandapplyingthesurfacedivergencetheoremonΓyields
d|u|2kdσ(x)≤−(2k−1)22−k|Γ|u|2k−1|2dσ(x)
dtΓΓ
−2k∂νu∙|u|2k−2udσ(x)+2kg(u)∙|u|2k−2udσ(x).
ΓΓAgainthisisjustifiedduetou∈X1.Addingtheseestimatesandobservingthatforeach
kitholds−(2k−1)22−k≤−2,weinfer
|u|L2k(Ω,RN)+|u|L2k(Γ,RN)≤−C||u|2|L22(Ω,RN)+|Γ|u|2|L22(Γ,RN)(5.5.4)
dkkk−1k−1
kkdt22
+2kf(u)∙|u|2−2udx+2kg(u)∙|u|2−2udσ(x).
ΓΩ

184

BoundaryConditionsofReactive-Diffusive-ConvectiveType

(II)Usingthesigncondition(5.5.3)andthat|ζ|2k−2≤|ζ|2k+1forζ∈RN,weestimate
theintegraltermsin(5.5.4)byaconstantmultipleof
2|u|L2k(Ω,RN)+|u|L2k(Γ,RN)+2.
k2k2kk
ItfollowsfromtheGagliardo-Nirenberginequality(PropositionA.6.1)andYoung’sin-
equalitythatforε∈(0,1)itholds
nn+21−nn+2−n/2
|v|L2(Ω,RN)≤C|v|W1(Ω,RN)|v|L1(Ω,RN)≤ε|v|W21(Ω,RN)+Cε|v|L1(Ω,RN).
2Fromthisinequalityweobtain
21−ε2−n/2−12
−|v|L2(Ω,RN)≤−|v|L2(Ω,RN)+Cε|v|L1(Ω,RN).
εNotethatthisestimateremainsvalidifonereplacesΩbyΓandnbyn−1,respectively.
Usingthatε−(n−1)/2−1≤ε−n/2−1forε∈(0,1)wemayestimatethegradienttermsin
yb(5.5.4)−C|u|Lk(Ω,RN)+|u|Lk(Γ,RN))+Cε||u||L1(Ω)+||u||L1(Γ).
1−ε2k2k−n/2−12k−122k−12
22εWethereforeobtainfrom(5.5.4)that
dt|u|L2k(Ω,RN)+|u|L2k(Γ,RN)≤C−ε+2|u|L2k(Ω,RN)+|u|L2k(Γ,RN))
d2k2k1−εk2k2k
+Cε−n/2−1||u|2|L2(Ω)+||u|2|L2(Γ)+C2k.
k−1k−1
11Nowwechooseε=δ2−kwithsmallδ>0suchthat
ε1−C−+2k≤−2k.
εWefurtherobservethat
||u||L1(Ω)+||u||L1(Γ)≤|u|L2k−1(Ω,RN)+|u|L2k−1(Γ,RN).
2k−122k−122k
settingTherefore,mk:=sup|u|L2k(Ω,RN)+|u|L2k(Γ,RN),k∈N0,
t∈[0,t+(u0))
wearriveattheestimate
dt|u|L2k(Ω,RN)+|u|L2k(Γ,RN)
d2k2k
≤−2|u|L2k(Ω,RN)+|u|L2k(Γ,RN)+C(2)mk−1+C2.(5.5.5)
k2k2kkn/2+12kk
(III)Nowsupposethatm0isfinite.ThentheGronwall’slemmayields,inductively,
|u|Lk(Ω,RN)+|u|Lk(Γ,RN)≤Cmax|u0|Lk(Ω,RN)+|u0|Lk(Γ,RN),2mk−1+1,
2k2k2k2kkn/22k
2222

5.5TheGlobalAttractorforSemilinearDissipativeSystems

185

andinparticularthateachnumbermkisfinite.Takingthe2k-throotsonbothsidesand
thesupremumovertontheleft-handsideweobtain
mk≤2sup|u|L2k(Ω,RN)+|u|L2k(Γ,RN)
2k2k1/2k
t∈[0,t+(u0))
kn/22k1/2k
≤Cmax|u0|L2k(Ω,RN)+|u0|L2k(Γ,RN),(2mk−1+1).
ThereisaconstantC,independentofk,suchthat
|u0|L2k(Ω,RN)+|u0|L2k(Γ,RN)≤C|u0|BC(Ω,RN).
Thusthesequence(mk)k∈Nsatisfiestherecursiveestimate
kkmk≤Cmax1,(2kn/2mk2−1+1)1/2,
withC≥1,andisthereforedominatedbythesequence(xk)k∈N,definedby
kx0=Cmax{m0,1},xk=(2kn/2)1/2xk−1,k∈N.
Since∞klimxk=Cx0(2kn/2)1/2=C2nmax{m0,1},
→∞k=1kobtainew|u(∙,u0)|BC([0,t+(u0))×Ω,RN)≤2limsupmk≤Cmax{m0,1}.
→∞kThisshowstheassertedestimateforanarbitraryinitialvalueu0∈X1.
(IV)Nowsupposethatu0∈X1isanequilibriumof(5.1.1).Using(5.5.5)directlyyields
mk≤C(2kn/2mk2k−1+1)1/2k≤Cmax2kn/2k+1mk−1,1,
wherenowsimplymk=|u0|L2k(Ω,RN)+|u0|L2k(Γ,RN).Asaboveweconcludethat
|u0|BC(Ω,RN)≤C2max|u0|L1(Ω,RN)+|u0|L1(Γ,RN),1,
andC2isindependentofu0sincetheconstantarisingin(5.5.5)isindependentofit.
TheabovelemmaandProposition5.5.2showthatfortheglobalexistenceofasolutionof
(5.5.1)itsufficestofindanaprioriL1bound,providedthereactiontermssatisfy(5.5.3).
WenowconsideraclassofreactiontermswheresuchanL1boundcaninparticularbe
obtained.Weassumethat(5.5.1)isconservative,i.e.,therearepotentialsF,G:RN→Rwith
−F=f,−G=g,F(0)=G(0)=0.
Wefurtherassumethat(5.5.1)isdissipative,inthefollowingsense.Therearenumbers
ci,di∈R,i=1,...,N,suchthat
f(ζ)g(ζ)
limsupi<ci,limsupi<di,i=1,...,N,(5.5.6)
|ζi|→∞ζi|ζi|→∞ζi

186

BoundaryConditionsofReactive-Diffusive-ConvectiveType

andthereisη>0suchthatfori=1,...,Nitholds
|ψ|L22(Ω,RN×n)+|Γψ|L22(Γ,RN×n)−2ci|ψ|L22(Ω,RN)−2di|ψ|L22(Γ,RN)
|ψ|L22(Ω,RN)+|ψ|L22(Γ,RN)≥η(5.5.7)
forallψ∈W21(Ω,RN)∩W21(Γ,RN).Observethat(5.5.7)isalwayssatisfiedforci,di<0.
Butitmayhappenthat(5.5.7)isvalidalthoughci>0anddi<0,orviceversa.In
thissensetheinterplaybetweenthereactiontermsinΩandonΓdeterminesif(5.5.1)is
dissipative,andthenon-dissipativenessofonereactiontermcanbecompensatedbythe
other.Thisisanalogoustothedissipativityconditionin[15]fornonlinearRobinboundary
conditions.Werecordsomesimpleconsequencesoftheaboveassumptions.
Lemma5.5.4.Assumethatfandgareconservativeanddissipative.Thenthereisa
constantc0∈Rsuchthat
fi(ζ)ζi≤ciζi2+c0,gi(ζ)ζi≤diζi2+c0,i=1,...,N.
Inparticular,fandgsatisfy(5.5.3).Moreover,forζ∈RNitholds
NNF(ζ)≥−c2iζi2−Nc0,G(ζ)≥−d2iζi2−Nc0.
=1i=1iProof.Thefirstassertionisclear.Forζ∈RNwesetζ=(0,ζ2,...,ζN)andcalculate
1F(ζ)=F(ζ)−f1(sζ1,ζ2,...,ζN)ζ1ds≥F(ζ)−c21ζ12−c0.
0IteratingthisargumentwiththeremainingN−1variablesyieldssecondassertion.
TheassumptionthatfandgareconservativeallowtoconstructaLyapunovfunctionfor
(5.5.1),whichalreadyappearedin[80].WedefineV:M→Rby
11
V(φ)=2|φ|2dx+F(φ)dx+2|Γφ|2dσ(x)+G(φ)dσ(x).
ΓΓΩΩNotethatViswell-definedandcontinuous,dueto
M→Wp1(Ω,RN)∩Wp1(Γ,RN)∩BC(Ω,RN).
Letu=u(∙,u0)bethesolutionof(5.5.1)withinitialvalueu0∈M,andt∈(0,t+(u0)).
Sinceu(t)∈X1wemayintegratebybypartstoobtain
dV(u(t))=−(Δu(t)+f(u(t)))∙∂tu(t)dx
tdΩ−Γ(ΔΓu(t)−∂νu(t)+g(u(t)))∙∂tu(t)dσ(x)
=−|∂tu(t)|L22(Ω,RN)−|∂tu(t)|L22(Γ,RN).(5.5.8)

(5.5.8)

5.5TheGlobalAttractorforSemilinearDissipativeSystems

187

HenceVisnonincreasingalongsolutionsof(5.5.1),anditisconstantonlyalongequilibria.
ThereforeVisastrictLyapunovfunctionfor(5.5.1).4
Lemma5.5.4andassumption(5.5.7)alsoallowtoobtainanenergyestimatefor(5.5.1),
asfollows.From(5.5.8)weobtainthatforu0∈Mandt∈(0,t+(u0))itholdsV(u(t))≤
V(u0),andfurthertheestimate
NV(φ)|φ|L22(Ω,RN×n)+|Γφ|L22(Γ,RN×n)−2ci|φ|L22(Ω,RN)−2di|φ|L22(Γ,RN)
=1i+|φ|L22(Ω,RN×n)+|Γφ|L22(Γ,RN×n)−1
|φ|2W21(Ω,RN)+|φ|2W21(Γ,RN)−1
holdstrue.Theaboveconsiderationsmaybesummarizedasfollows.
Lemma5.5.5.Supposethat(5.5.1)isconservativeanddissipative,andletp∈(n+2,∞).
ThenV:M→RisastrictLyapunovfunctionfor(5.5.1),andthereisC>0suchthat
foreachu0∈Mthesolutionu(∙,u0)of(5.5.1)satisfies
+sup|u(t,u0)|2W21(Ω,RN)∩W21(Γ,RN)≤C(1+V(u0)).
t∈[0,t(u0))
Weusetheaboveaprioriestimatetoshowthat(5.5.1)generatesacompactglobalsemiflow
inM,withrelativelycompactorbits.
Proposition5.5.6.Supposethat(5.5.1)isconservativeanddissipative,andletp∈(n+
2,∞).Thenforu0∈Mthecorrespondingsolutionu(∙,u0)existsglobally,t+(u0)=+∞,
andtheorbit{u(t,u0)}t∈[0,∞)isrelativelycompactinM.Moreover,foreacht>0the
solutionmapu(t,∙):M→Miscompact.
Proof.(I)TheLemmas5.5.3and5.5.5yieldthatu(∙,u0)isboundedinBC(Ω,RN),and
thusProposition5.5.2yieldst+(u0)=+∞.Thecompactnessofthetime-t-mapu(t,∙):
M→Mforallt>0followsfromTheorem5.3.3.
(II)FortherelativecompactnessoforbitswealsowanttoapplyProposition5.5.2,and
thereforehavetoshowthat{u(t,u0)}t∈[0,∞)isboundedinMpsforsomes>0.Dueto
Sobolev’sembeddingsitholds
Wrσ(Ω,RN)→Wqs(Ω,RN)forσ−n≥s−n,σ≥s,r≥q,
qrandthisremainstrueifonereplacesΩbyΓandnbyn−1,respectively.ByLemma5.5.5
thesolutionisboundedintheenergyspacesW21(Ω,RN)andW21(Γ,RN).Wetherefore
inoundednessbtheobtainWqs(Ω,RN),s=1−n/2+n/q,andWqs+1−1/q(Γ,RN),s=1/2−n/2+n/q,
whereq<n2−n1.Thusfortheseqthereisasmalls>0suchthattheorbitisboundedin
Mqs=Wqs(Ω,RN)∩Wqs+1−1/q(Γ,RN).
4IntheliteratureitissometimesrequiredthataLyapunovfunctionisboundedfrombelow.Inthe
contextofcompactsemiflowsthispropertyisnotnecessary,cf.[16,Remark1.1.4].

188

BoundaryConditionsofReactive-Diffusive-ConvectiveType

ByLemma5.5.1,theorbitisthereforeboundedinMq2−2/q.Ifq>n+1thenSobolev’s
embeddingsyieldM2q−2/q→Mpsforsomes>0,andwearedone.2Otherwiseweiterate
theapplicationofSobolev’sembeddingsandLemma5.5.5asintheproofofProposition
4.4.2,toobtaintheboundednessoftheorbitinMpsforsomes>0afterfinitelymany
steps.Thelaststeptowardstheglobalattractorfor(5.5.1)istheboundednessofthesetofits
equilibria.Lemma5.5.7.Supposethat(5.5.1)isconservativeanddissipative,andletp∈(1,∞).
ThenthesetofitsequilibriaisboundedinX1.
Proof.(I)Anequilibriumv∈X1→Wp2(Ω,RN)∩Wp3−1/p(Γ,RN)of(5.5.1)solves
Δv+f(v)=0inΩ,
ΔΓv−∂νv+g(v)=0onΓ.
Multiplyingthedomainequationwithv,integratingoverΩ,integratingbypartsandusing
obtainew5.5.4LemmaN|v|L22(Ω,RN×n)−Γ∂νv∙vdσ(x)−ci|vi|L22(Ω,RN)−Nc0≤0.
=1iEmployingtheboundaryequationyieldsinasimilarwaythat
N|Γv|L22(Γ,RN×n)+Γ∂νv∙vdσ(x)−di|vi|L22(Γ,RN)−Nc0≤0.
=1iAddingtheseestimatesweobtain,using(5.5.7),
N2Nc0≥|v|L22(Ω,RN×n)+|Γv|L22(Γ,RN×n)−ci|vi|L22(Ω,RN)+di|vi|L22(Γ,RN)
=1i|v|2W21(Ω,RN)+|v|2W21(Γ,RN).
Thisestimate,togetherwithLemma5.5.3,leadsto
sup{|v|BC(Ω,RN):v∈X1isequilibriumof(5.5.1)}<∞.(5.5.9)
(II)Wehaveseenthattherealizationof
0Δ−A=−∂νΔΓ,D(−A)=X1,
onX0=Lp(Ω,RN)×Wp1−1/p(Γ,RN)isthegeneratorofananalyticC0-semigroup.In
particular,thereisλ>0suchthatA+λisanisomorphismX1→X0.Foranequilibrium
vof(5.5.1)wemaythereforeestimate
|v|X1=|(A+λ)−1(f(v)+λv,g(v)+λv)|X1
|f(v)+λv|Lp(Ω,RN)+|g(v)+λv|Wp1−1/p(Γ,RN)

5.5TheGlobalAttractorforSemilinearDissipativeSystems

Itfollowsfrom(5.4.6)that
|g(v)|Wp1−1/p(Γ,RN)≤C|v|BC(Ω,RN)1+|v|Wp1−1/p(Γ,RN).
Using(5.5.9),theinterpolationinequalityandYoung’sinequalitywethusobtain

189

|v|X11+|v|Wp1−1/p(Γ,RN)≤ε|v|Wp3−1/p(Γ,RN)+Cε,
whereε>0isarbitrary.Choosingεappropriately,wemaysubtract21|v|Wp3−1/p(Γ,RN)on
bothsidesoftheaboveinequality,whichyieldsauniversalX1-boundfortheequilibriaof
(5.5.1).

Theconsiderationsinthissection,togetherwith[63,Theorem2.3],yieldthefollowing
resultonthelong-timebehaviourof(5.5.1).

Theorem5.5.8.Supposethat(5.5.1)isconservativeanddissipative.Thenitgeneratesa
compactglobalsemiflowofsolutionsinthephasespaceM,andthesetEofitsequilibria
isnonempty.ThesemiflowpossessesaconnectedglobalattractorA⊂X1,theω-limitset
ofeachorbitiscontainedinE,andalsotheα-limitsetofeachcompleteorbitiscontained
inE.IfEisdiscrete,thenAconsistspreciselyofequilibriaandcompleteorbitsconnecting
them.Ifinadditionfandgdonothaveacommonzero,then(5.5.1)hasatleastone
nonconstantequilibrium,andeachsolutionconvergesinMtosuchapattern.

AendixApp

endixApp

A.1BoundariesofDomainsinRn
LetΩ⊂Rnbeadomainwithboundary∂Ω,n≥2.Wesaythat∂Ωissmooth,ifforeach
x∈∂ΩthereareaboundedopensetU⊂Rnwithx∈Uandasmoothdiffeomorphism
ϕ:U→Rnwith
ϕ(U∩Ω)⊂R+n,ϕ(U∩∂Ω)⊂Rn−1×{0}=∼Rn−1.
Thepair(U,ϕ)iscalledachartfor∂Ωaroundx.Theparametrizationg:Rn−1∩ϕ(U)→
∂Ωof∂Ωaroundxwithrespectto(U,ϕ),alsocalledlocalcoordinates,isdefinedby
g(y):=ϕ−1(y,0),y∈Rn−1∩ϕ(U).
Itholdsthatg:Rn−1∩ϕ(U)→∂Ω∩Uisahomeomorphism,andthatthederivative
g(y)∈B(Rn−1,Rn)hasmaximalrankn−1foreachy∈Rn−1∩ϕ(U).
ThetangentialspaceTx∂Ωon∂Ωinxisgivenbytheimageofthematrixg(ϕ(x)),andit
hasthedimensionn−1.ItbecomesaHilbertspacewhenconsideringitasaclosedsubspace
ofRn.AcanonicalbasisofTx∂Ωisgivenby{∂1g(ϕ(x)),...,∂n−1g(ϕ(x))}.Theouterunit
normalν(x)∈Rnon∂Ωatxisgivenanormalizedelementoftheorthogonalcomplement
ofTx∂ΩinRn.Thetangentialspacesandtheouterunitnormalsareindependentofthe
chartandthecorrespondingparametrization,andtheouterunitnormalfieldissmooth.

ThefundamentalformG=(gij)i,j=1,...,n−1withrespectto(U,ϕ)isdefinedby
gij:=∂ig∙∂jg,i,j=1,...,n−1,
wherethescalarproductistakeninRn,andG(y)isforally∈Rn−1∩ϕ(U)asymmetric
positivedefinitematrix.ItsinverseG−1:=(gij)i,j=1,...,n−1isalsosymmetricandpositive
definite.Thedeterminant|G|ofGiscalledtheGramiandeterminant.
Itisusefultohavechartsandparametrizationswiththefollowingspecialproperties.

192

endixApp

LemmaA.1.1.LetΩ⊂RNhaveasmoothboundary∂Ω,letx∗∈∂Ω,r>0andlet
Oν(x∗)beanyorthogonalmatrixthatrotatesν(x∗)to(0,...,0,−1)∈Rn.Thenthereisa
chart(U,ϕ)for∂Ωaroundx∗withtheproperties
ϕ(x∗)=0,ϕ(x∗)=Oν(x∗),ϕ(U)=Br(0),ϕ(U∩Ω)⊂R+n,ϕ(U∩∂Ω)⊂Rn−1,
andthecorrespondingfirstfundamentalformsatisfiesG(x∗)=idRn−1.Thecoordinatesg
tosuchachartarecalledassociatedtothepointx∗∈∂Ω.
Proof.Thediffeomorphismx→Oν(x∗)(x−x∗)translatesx∗intotheoriginandrotates
Tx∂ΩtoRn−1×{0}.TheimplicitfunctiontheoremimpliesthatOν(x∗)(∂Ω−x∗)maylocally
aroundtheoriginberepresentedasagraphofsmoothfunctionh:U→Rwithh(0)=0,
whereU⊂Rn−1isopen,suchthat∂Ωlieslocallyintheset{y=(y,yn):yn=h(y)}.
Settingy=Oν(x∗)(x−x∗),weobtainthatϕ(x):=(y,yn−h(y))definesachartfor∂Ω
aroundx∗,whichhasthedesiredpropertiesafterrestrictiontothepreimageofBr(0).
IfEisaBanachspaceand(U,ϕ)achartfor∂Ωonedefinesthepush-forwardoperatorΦ
forfunctionsu:Ω∩U→Eby
Φu:Rn+∩ϕ(U)→E,Φu:=u◦ϕ−1.
Similarly,onedefinesthepull-backoperatorΦ−1forfunctionsv:R+n∩ϕ(U)→Eby
Φ−1v:Ω∩U→E,Φ−1v=v◦ϕ.
Itisshownin[86,Thm.10.3]thattheprincipalpartofadifferentialoperatorfortunately
transformsinasimpleway.
LemmaA.1.2.LetEbeaBanachspace,letP(x,)=|α|≤kpα(x)αbeadifferential
operatoroforderk∈N0onΩwithpα(x)∈B(E),andlet(U,ϕ)beachartfor∂Ω.Then
fortheprincipalpartofthetransformedoperatorPΦ,whichisforv:R+n∩ϕ(U)→E
ybengivPΦ(x,)v:=ΦP(∙,)Φ−1v(x),x∈R+n∩ϕ(U),
itholdsPΦ(x,)=P(ϕ−1(x),ϕ(x)T).
Finally,assumethat∂Ωiscompact.Thenthereisfinitecollectionofcharts(Ui,ϕi)with
∂Ω⊂iUi.
Thereisfurtherasmoothpartitionofunity{ψi}for∂ΩsubordinatetothecoveriUi,
i.e.,suppψi⊂Uiforalli.
TheoryolationterpInA.2dLet(E0,|∙|E0)and(E1,|∙|E1)beBanachspaceswithE1→E0.Forθ∈(0,1)and
p∈[1,∞]therealinterpolationspaces(E0,E1)θ,pandthecomplexinterpolationspaces
[E0,E1]θaredefinedandinvestigatedin[13,68,82].Welistsomewell-knownpropertiesof

ryTheoolationInterpA.2

193

thesespaces.RecallthatbytheequalityofBanachspaceswemeantheycoincideassets
andhaveequivalentnorms.
letThroughout,0<θ<1,0<θ1<θ2<1,p∈[1,∞].
Thenthefollowingholdstrue.
a)For1≤p1≤p2≤∞:E1→d(E0,E1)θ,p1→d(E0,E1)θ,p2→dE0,see[68,Prop.1.3,
1.17].b)E1→d[E0,E1]θ2→d[E0,E1]θ1→dE0,see[82,Thm.1.9.3].
c)Forq∈[1,∞](see[13,Thm.4.2.1,4.7.1]):
(E0,E1)θ2,p→d(E0,E1)θ1,q,(E0,E1)θ2,p→d[E0,E1]θ1,[E0,E1]θ2→d(E0,E1)θ1,p.
d)IfF1isaBanachspacewithE1→dF1→dE0:
(E0,E1)θ,p→(E0,F1)θ,p,[E0,E1]θ→[E0,F1]θ.
e)Forq1,q2∈[1,∞](see[13,Thm.3.5.3]):
(E0,E1)θ1,q1,(E0,E1)θ2,q2θ,p=(E0,E1)(1−θ)θ1+θθ2,p.
f)[E0,E1]θ1,[E0,E1]θ2θ=[E0,E1](1−θ)θ1+θθ2,
andthisassertionremainsvalidif[E0,E1]θ1isreplacedbyE0andθ1=0or[E0,E1]θ2
isreplacedbyE1andθ2=0(see[82,Rem.1.9.3/1]).
g)[E0,E1]θ1,[E0,E1]θ2θ,p=(E0,E1)(1−θ)θ1+θθ2,p,
andthisassertionremainsvalidforθ1=0orθ2=0asine)(see[82,Thm.1.10.3/2]).
h)IfE0andE1arereflexive(see[82,Rem.1.10.3/2]):
(E0,E1)θ1,p,(E0,E1)θ2,pθ=(E0,E1)(1−θ)θ1+θθ2,p.
i)IfF1→dF0withF1→dE1,F0→dE0,areBanachspaces(see[68,Thm.1.6,2.6]):
B(E0,F0)∩B(E1,F1)→B(E0,F0)θ,p,(E1,F1)θ,p∩B[E0,F0]θ,[E1,F1]θ.
Moreprecisely,forA∈B(E0,F0)∩B(E1,F1)itholds
|A|B((E0,F0)θ,p,(E1,F1)θ,p)≤|A|B1(−Eθ0,F0)|A|Bθ(E1,F1),
andanalogouslyfor|A|B([E0,F0]θ,[E1,F1]θ).
j)Bytheinterpolationinequality(see[68,Cor.1.7,2.8]):
1−θθ1−θθ
|x|(E0,E1)θ,p≤C(θ,p)|x|E0|x|E1,|x|[E0,E1]θ≤|x|E0|x|E1x∈E1.

endixApp

endixApp194k)IfA,D(A)isthegeneratorofaboundedC0-semigroup{T(t)}t≥0onE0,andD(A)
isequippedwiththegraphnorm(see[68,Prop.5.7]):
∞E0,D(A)θ,p=u∈E0:[u]∗p:=t−θp|T(t)u−u|pE0dt<∞,
t0wherethespaceontheright-handsideisequippedwiththenorm|u|E0+[u]∗.
l)IfA,D(A)isthegeneratorofaboundedanalyticC0-semigroup{T(t)}t≥0onE0,
andD(A)isequippedwiththegraphnorm(see[82,Thm.1.14.5]):
∞E0,D(A)θ,p=u∈E0:[u]p∗∗:=tp(1−θ)|AT(t)u|Ep0dtt<∞,
0wherethespaceontheright-handsideisequippedwiththenorm|u|E0+[u]∗∗.In
[u]∗∗onemayrestricttheintegrationovertto(0,δ),δ>0.If{T(t)}t≥0isinaddition
exponentiallystablethenitsufficestotake[u]∗∗asanorm.
m)If(Ω,ν)isaσ-finitemeasurespaceandθ∈(0,1),p∈[1,∞)(see[82,Thm.1.18.4]):
Lp(Ω;E0),Lp(Ω;E1)θ,p=LpΩ;(E0,E1)θ,p,
Lp(Ω;E0),Lp(Ω;E1)θ=LpΩ;[E0,E1]θ.
n)If(Ω,ν)isaσ-finitemeasurespaceand1≤p1<p2≤∞(see[82,Thm.1.18.4]):
Lp1(Ω;E),Lp2(Ω;E)θ=Lq(Ω;E),where1=1−θ+θ.
ppq21Hereoneinterpolatesinfactbetweenaninterpolationcouple,cf.[13,68,82].
ThefollowingHardy-Younginequalitiesareusefulforinterpolationtheory.Itholds
p
Tt−αtu(s)dsdt≤1Tt−αu(t)pdt,(A.2.1)
00stαp0t
p
TtαTu(s)dsdt≤1pTtαu(t)pdt,(A.2.2)
0tstα0t
forallnonnegativemeasurablefunctionsu:(0,T)→R,T∈(0,∞],allα>0andall
p∈[1,∞),cf.[50,p.245-246].
eratorsOpSectorialA.3Fordetailedinformationsontheconceptsdescribedinthissectionwereferto[7,24,48,
55,62,68,82]andthereferencestherein.Throughout,letEbeacomplexBanachspace.
ThespaceEissaidtobeofclassHTiftheHilberttransformonL2(R;E)isbounded,
i.e.,ifthedenselydefinedoperatorH,givenby
1ϕ(s−t)
(Hϕ)(s):=ε→lim0π|t|≥εtdt,ϕ∈S(R;E),

A.3SectorialOperators

195

mayuniquelybeextendedtoL2(R;E).Thispropertyisequivalenttotheboundednessof
theoperatorcorrespondingtothesymbol−isign.ItisfurtherequivalentforEtohave
thepropertyofunconditionalmartingaledifferences,animportantconceptinstochastic
analysis.ThereforespacesofclassHTarealsocalledUMD-spacesintheliterature.Since
wehaveapurelyanalyticpointofviewinthiswork,wepreferthefirstnotion.
BanachspacesofclassHTarealwaysreflexive.Manyspacesinapplicationsareofclass
HT,likefinitedimensionalspaces,Hilbertspaces,andfurtherLp-,Sobolev,Slobodetskii,
BesovandBesselpotentialspacesinthereflexiverange,providedtheytakevaluesina
spaceofclassHT(seeAppendixA.4).L1-,L∞andCk-spacesarenotofclassHT.The
HT-propertyisstableunderrealandcomplexinterpolation.Formoreinformationand
proofswereferto[7,SectionsIII.4.3-4.5]and[55,Chapters6-8].
AfamilyofboundedoperatorsT⊂B(E0,E1)betweenBanachspacesE0,E1iscalled
R-boundedifthereisC>0suchthat,forallT1,...,Tm∈Tandx1,...,xm∈E0with
m∈N,itholdsmm
rnTnxnL2(0,1;E1)≤CrnxnL2(0,1;E0),
=1n=1nwherern(t):=signsin(2nπt)denotetheRademacherfunctionson[0,1].Westressthat
thenormsareoutsidethesums.IfE0,E1areHilbertspacesthisnotionisequivalentto
theuniformboundednessofT.TheinfimumofallCsatisfyingtheaboveestimateiscalled
theR-boundofTandisdenotedbyR(T).Wereferto[24,Chapter3],[62,Chapter2]
and[55,Chapter4]fordetailedinformations.
Roughlyspeaking,spacesofclassHTandtheconceptofR-boundednesscanbeusedto
showtheboundednessofoperatorsforwhichstandardnormestimatesdoactuallynot
leadtoboundedness.Theadvantageisthatonecanavoidtheapplicationofthetriangle
inequalityatcertainpoints,and’leavethenormoutsideasum’whileestimating.Besides
theinterestintheirown,thecombinationoftheseconceptsleadstoimportantresultsfor
theapplicationstopartialdifferentialequations,liketheoperator-valuedFourier-multiplier
theoremduetoWeis[85],thejointfunctionalcalculusduetoKaltonandWeis[62],the
characterizationofmaximalLp-regularity[85],theDore-Vennitheorem[31],andmany
more(seebelowformoredetailsonthelattertworesults).
LetAbeanoperatoronEwithdomainD(A).WecallAsectorialifAisclosed,densely
defined,hasdenserangeandifitholds(−∞,0)⊂ρ(A)withtheresolventestimate
|t(t+A)−1|B(E)≤C,t>0,
forsomeC>0.Definetheopensector
Σθ:=λ∈C\{0}:|argλ|<θ.
IfAissectorial,thentheresolventestimateandaNeumannseriesyieldanangleφ∈[0,π)
suchthatΣπ−φ⊂ρ(−A).OnemaythusdefinethespectralangleofAby
φA:=infφ∈[0,π):Σπ−φ⊂ρ(−A),sup|λ(λ+A)−1|<∞.
λ∈Σπ−φ

196

endixApp

AsectorialoperatorAisthegeneratorofaboundedanalyticC0-semigroupifandonly
ifφA<π/2([35,TheoremII.4.6]).ForφA=π/2theoperatorAdoesnotnecessarily
generateasemigroup.Notethatsometimesintheliteratureonlygeneratorsofanalytic
sectorial.dcallearesemigroupsTheoperatorAiscalledR-sectorialifitissectorialandifthefamilyofoperators
t(t+A)−1:t>0
isR-boundedinB(E).AsaboveonemaydefinetheR-angleφARofAby
φAR:=infφ∈(φA,π):R{λ(λ+A)−1:λ∈Σπ−φ}<∞.
ItisclearfromthedefinitionsthatφA≤φAR.Theimportanceofthisconceptliesinthe
factthatonaBanachspaceofclassHTthegeneratorofanexponentiallystableC0-
semigroupAisR-sectorialwithφAR<π/2ifandonlyifitenjoysthepropertyofmaximal
Lp-regularityonthehalflineforp∈(1,∞),i.e.,ifforeachf∈Lp(R+;E)thereisaunique
solutionu∈Wp1(R+;E)∩Lp(R+;D(A))oftheproblem
u+Au=f,t>0,u(0)=0.
ThisresultisduetoWeis[85].Formoreinformationson(R-)sectorialoperatorswerefer
to[24,62],andto[30]forasurveyonmaximalLp-regularity(seealsoSection1.2.1).
Wenowconsiderthefunctionalcalculusforsectorialoperators.Forφ∈(0,π)onedefines
rasalgebfunctiontheH(Σφ):=f:Σφ→C:fisholomorphic,
H∞(Σφ):=f:Σφ→C:fisholomorphicandbounded,
H0(Σφ):=f∈H(Σφ):thereareα,β>0with|λsup|≤1|λ−αf(λ)|+|λsup|≥1|λβf(λ)|<∞,
|λ|≤1|λ|≥1
H1(Σφ):=f∈H(Σφ):thereareα,β∈Rwithsup|λ−αf(λ)|+sup|λβf(λ)|<∞.
NowfixasectorialoperatorA,andletφ∈(φA,π].ForacurveΓ=(∞,0]eiψ∪[0,∞)e−iψ
withψ∈(φA,φ)themap
ΦA:H0(Σφ)→B(E),f(A):=ΦA(f):=1f(λ)(λ+A)−1dλ,
iπ2ΓdefinesafunctionalcalculusforA,i.e.,analgebrahomomorphism.Ifthereareφ∈(φA,π)
andaconstantKφsuchthat
|f(A)|B(E)≤Kφ|f|∞,f∈H0(Σφ),
thenthefunctionalcalculusforAmayuniquelybeextendedfromH0(Σφ)toH(Σφ).In
thiscaseAissaidtoadmitaboundedH∞-calculus,andtheH∞-angleφA∞ofAisdefined
astheinfimumofallφ∈(φA,π)thatadmitanestimateasabove.

A.3SectorialOperators

197

Forf∈H1(Σφ)andψ(λ):=1+λλ2,choosek∈Nsuchthatψkf∈H0(Σφ).Thenthe
ratoreopf(A):=ψ(A)−k(ψkf)(A),D(f(A))=x∈E:(ψkf)(A)x∈D(Ak)∩R(Ak),
isclosedanddenselydefinedonE,anditisindependentofthechosennumberk.Themap
f→f(A)iscalledtheextendedfunctionalcalculusforA.Wereferto[24,48]forproofs
andmoreinformationsonthefunctionalcalculusforsectorialoperators.
Forz∈Ctheextendedfunctionalcalculusinparticularallowstodefinethefractional
powersAzofasectorialoperatorA.Ingeneral,Azisaclosedanddenselydefinedoperator
onX,andthedomainssatisfy
D(Az1)→dD(Az2)→dE,Rez1>Rez2>0,
whenequippedwiththegraphnorm,cf.[24,Thm.2.1],[68,Lem.4.11]and[7,Thm.
III.4.6.5].IfAisinvertiblethenAzisaboundedoperatorforRez<0,andforRez,Reω>
0theoperatorAzisanisomorphism
D(Az+ω)→D(Aω),D(Az)→E,
seeagain[7,Thm.III.4.6.5].Forp∈(1,∞)andθ,Rez>0withθ+Rez<1,theoperator
Azisfurtheranisomorphism
(E,D(A))Rez+θ,p→(E,D(A))θ,p,
cf.[82,Thm.1.15.2].Thereiterationtheorem[48,Prop.6.6.7]statesthatforθ∈(0,1)
andp∈(1,∞)itholds
(E,D(Az1))θReRezz12,p=(E,D(Az2))θ,p,0<Rez2<Rez1,
ervmoreoandzωz(E,D(A1))(1−θ)ReRezω1+θReRezz12,p=(D(A),D(A2))θ,p,0<Reω<Rez2<Rez1.
Fors=k+θ≥0withk∈N0,p∈[1,∞]andθ∈[0,1)itisconvenienttodefine
DA(s,p):=D(As)ifs∈N0,
DA(s,p):=x∈D(Ak):Akx∈(E,D(A))θ,pifs∈/N0,
wherethesespacesareequippedwiththenorm|x|E0+|Akx|(E,D(A))θ,p,respectively(with
E=(E,D(A))0,p).
Therearerulesanalogouslytotheonesforpowersofscalars.Itholdsthat
AzAω⊂Az+ω,D(Aω)∩D(Az+ω)=D(AzAω),z,ω∈C,

198

endixApp

see[48,Prop.3.2.1],andforRez,Reω>0wehaveAzAω=Az+ωby[48,Prop.3.1.1].
Forα∈Rwith|α|<π/φAtheoperatorAαissectorial,anditholdsφAα=|α|φAand
(Aα)z=Aαz,z∈C,
cf.[48,Prop.3.1.4,Cor.3.1.5].
ClosedlinearoperatorsA,BonEarecalledresolventcommutingifthereareλ∈ρ(A),
µ∈ρ(B)suchthat
(λ−A)−1(µ−B)−1=(µ−B)−1(λ−A)−1.
Forsuchoperatorsrealinterpolationcommuteswiththeintersectionofthedomains[47].
LemmaA.3.1.LetAandBberesolventcommutingsectorialoperatorsonaBanach
spaceE.Thenforθ∈(0,1)andp∈(1,∞)itholds
(E,D(A)∩D(B))θ,p=(E,D(A))θ,p∩(E,D(B))θ,p.
AsectorialoperatorAissaidtoadmitboundedimaginarypowersifAis∈B(E)forall
s∈Randifthereareε>0andC>0with|Ais|B(E)≤Cfor|s|≤ε.Inthiscase{Ais}s∈R
formsaC0-groupofboundedoperatorsonE,andthegrowthboundθAofthisgroup,i.e.,
1θA=|slim|→∞sup|s|log|Ais|B(E),
iscalledthepowerangleofA.
Operatorswithboundedimaginarypowersenjoyverygoodproperties.IfAisinvertible
andadmitsboundedimaginarypowersthenYagi’stheoremstatesthatfor0≤Reω<Rez
andθ∈(0,1)itholds
[D(Aω),D(Az)]θ=D(A(1−θ)ω+θz),(A.3.1)
wherethedomainsareagainequippedwiththegraphnorm,respectively[82,Thm.1.15.3].
Theaboveidentityisusefultodetermineacomplexinterpolationspaceinconcretesitua-
tions.ThefollowingresultisavariantoftheDore-Vennitheorem[31,72].
TheoremA.3.2.LetEbeofclassHT,andsupposethattheoperatorsA,Bareresolvent
commutingandadmitboundedimaginarypowerswithθA+θB<π.Thenforallρ>0
true.holdswingfollothea)A+ρBwithD(A+ρB)=D(A)∩D(B)isclosedandsectorial;
b)A+ρBadmitsboundedimaginarypowerswithθA+ρB≤max{θA,θB};
c)thereisaconstantC>0,independentofρ,suchthat
|Ax|E+ρ|Bx|E≤C|Ax+ρBx|E,x∈D(A)∩D(B).
IfAorBisinvertible,thenA+ρBisinvertibleaswell.

A.3SectorialOperators

199

Thenextresultisalsocalledthemixedderivativetheorem,see[78]andalso[27].
LemmaA.3.3.InthesituationoftheDore-VenniTheoremA.3.2,foreachα∈[0,1]it
holdsB1−αx∈D(Aα)forx∈D(A)∩D(B),andthereisaconstantC>0suchthat
|AαB1−αx|E≤C|Ax+Bx|Eforallx∈D(A)∩D(B).
Inparticular,B1−α∈B(D(A)∩D(B),D(Aα)).
WenextstateavariantofLemmaA.3.1forcomplexinterpolation,see[37,Lem.9.5].
LemmaA.3.4.LetEbeofclassHTandθ∈(0,1).SupposethattheoperatorsA,Bare
resolventcommuting,thattheyadmitboundaryimaginarypowerswithθA+θB<π,and
thatAorBisinvertible.Then
[E,D(A+B)]θ=[E,D(A)]θ∩[E,D(B)]θ,θ∈(0,1),
i.e.,D((A+B)θ)=D(Aθ)∩D(Bθ).
Formorepropertiesofoperatorswithboundedimaginarypowerswereferto[7,Sec.III.4.7],
[68,Sec.4.2]and[24,Sec.2.3].
TheabovepropertiesofasectorialoperatorAonaBanachspaceEofclassHTarerelated
asfollows.IfAadmitsaboundedH∞-calculusthenAadmitsboundedimaginarypowers,
andthelatterpropertyimpliesthatAisR-sectorial[24,Sec.2.4,Thm.4.5].Theangles
satisfyφA∞≥θA≥φAR≥φA.(A.3.2)
Inparticular,ifEisofclassHTandAadmitsaboundedH∞-calculusorboundedimag-
inarypowerswithanglesstrictlysmallerthanπ/2,thenAenjoysmaximalLp-regularity
onthehalf-lineforallp∈(1,∞).Theconverseoftheaboveassertionsisfalse,ingeneral.
ThestandardexamplesforoperatorswithaboundedH∞-calculusareforp∈(1,∞)and
aBanachspaceofclassHTthederivative∂tonLp(R;E)withdomainWp1(R;E),andthe
negativeLaplacian−ΔnonLp(Rn;E)withdomainWp2(Rn;E).Fortheangleswehave
φ∂∞t=π/2andφ−∞Δn=0.Foraproofwereferto[24,Thm.5.5]and[48,Ch.8](seealso
Theorem1.1.7andLemma1.3.1).
Wenowconsiderfurtherpropertiesofsectorialoperators.Wealreadysawthatareal
fractionalpowerofasectorialoperatorremainssectorialifthepowerandthesectoriality
angleareappropriate.Asimilarresultistrueforotherpropertiesofanoperator.
LemmaA.3.5.AssumethatAadmitsaboundedH∞-calculusorboundedimaginary
powers,andletα>0satisfy
α<π/φA∞,orα<π/θA.
ThenAαenjoysthesameproperty,with
φA∞α≤αφA∞,orθAα≤αθA.

200

endixApp

Proof.(I)FirstsupposethatAadmitsaboundedH∞-calculus.Takeasmallε>0with
α(φA∞+ε)<πandφA∞+ε<π.Thenforf∈H0∞(Σφ)withφ∈αφA∞,α(φA∞+ε)the
functionλ→f(λ):=f(λα)belongstoH0∞(Σφ/α),andφ/α∈(φA∞,π).UsingthatAα
issectorialandthecompositionruleforthefunctionalcalculusofsectorialoperators[48,
Theorem2.4.2]weobtain
|f(Aα)|B(E)=|f(A)|B(E)≤Kφ/α|f|∞=Kφ/α|f|∞.
HenceAαadmitsaboundedH∞-calculusofanglenotlargerthanαφA∞.
(II)NowassumethatAadmitsboundedimaginarypowers.Then(Aα)is=Aisα∈B(E)
foralls∈R,and|(Aα)is|≤Cforall|s|≤ε/α.ThusAαalsoadmitsboundedimaginary
powers.Moreover,dueto[7,CorollaryIII.4.7.2],forallθ>θAwehave|Aisα|B(E)≤Ceθα|s|.
TakingthelogarithmyieldsthatθAα≤αθ,andtheassertionfollows.
Foraσ-finitemeasurespace(Ω,ν)andp∈(1,∞)wemaydefinethepointwiserealization
ofAponLp(Ω;E)by
(Apu)(t):=Au(t),a.e.t∈Ω,u∈D(Ap):=Lp(Ω;D(A)),
whereD(A)isendowedwiththegraphnorm.WeshowthatApenjoysthesameproperties
.AasLemmaA.3.6.Let(Ω,ν)beaσ-finitemeasurespace,andsupposethattheoperatorAis
sectorial,admitsaboundedH∞-calculusoradmitsboundedimaginarypowers.Thenfor
p∈(1,∞)thepointwiserealizationApofAonLp(Ω,E)enjoysthesameproperty,with
φAp≤φA,φA∞p≤φA∞,orθAp≤θA.
Inaddition,ifAissectorialandf∈H1∞(Σφ)withφ∈(φA,π),thenf(Ap)=f(A)p.
Proof.(I)SupposethatAissectorial.WefirstshowthatApisdenselydefined.Let
ε>0begiven,andletim=1αixi∈Lp(Ω;E)beastepfunction,withm∈N,αi∈
Lp(Ω),αi=0andxi∈E.SinceD(A)isdenseinEwefindyi∈D(A)with|xi−
yi|E≤ε/(m|αi|Lp(Ω))−1.Itthenholdsim=1αiyi∈D(Ap)=Lp(Ω;D(A)),andfurther
|im=1αixi−im=1αiyi|Lp(Ω;E)<ε.SincethestepfunctionsaredenseinLp(Ω;E),it
followsthatD(Ap)isdenseinLp(Ω;E).ThedensityoftherangeofApinLp(Ω;E)is
showninasimilarway.
Nowletλ∈ρ(A).Thenforh∈Lp(Ω;E)theuniquesolutionu∈Lp(Ω;E)of(λ+Ap)u=h
isforalmosteveryt∈Ωgivenbyu(t)=(λ+A)−1h(t).HenceApisclosed,anditholds
ρ(A)⊂ρ(Ap),(λ+Ap)−1=(λ+A)−1pforλ∈ρ(A).(A.3.3)
Thisyieldsforλ∈ρ(A)theestimate
|λ(λ+Ap)−1|B(Lp(Ω;E))=sup|λ(λ+Ap)−1h|Lp(Ω;E)≤|λ(λ+A)−1|B(E),
|h|Lp(Ω;E)=1
whichshowsthatApissectorialwithφAp≤φA.

A.4FunctionSpacesonDomainsandBoundaries

201

(II)IfAissectorialthenweinferfrom(A.3.3)thatforf∈H0∞(Σφ)withφ∈(φA,π),
h∈Lp(Ω;E)andalmosteveryt∈Ωitholds
f(Ap)h(t)=21πiΓf(λ)(λ+Ap)−1h(t)dλ=f(A)ph(t),
whichyieldsf(Ap)=f(A)p.Similarlyoneobtainsthisidentityforf∈H1∞(Σφ).Using
thisfactandestimatingasaboveitisstraightforwardtocheckthattheotherproperties
ofAcarryovertoApasasserted.

A.4FunctionSpacesonDomainsandBoundaries
WefirstconsiderfunctionspaceswithvaluesinaBanachspaceofclassHT,andreferto
[8,9,75,91]formoredetailsandproofs.Forscalar-valuedfunctionspaceswereferto[82].
TheHT-valuedfunctionspacessharemanypropertieswiththescalar-valuedspaces,due
tothefactthatappropriateFouriermultipliertheoremsareavailable.
LetEbeacomplexBanachspaceofclassHT,andletΩ⊂Rnbeadomainwithsmooth
boundary.Wedenoteby=(∂1,...,∂n)theeuclidiangradientonΩ,andα∈N0ndenotes
amultiindex.Fork∈N0theBanachspaceoftheE-valuedk-timesboundeduniformly
continuouslydifferentiablefunctionsonΩisdenotedby
BUCk(Ω;E),
equippedwithitscanonicalnorm.Fors=[s]+s∗∈R+\Nwith[s]∈N0ands∗∈[0,1)
theBanachspaceofboundedHöldercontinuousfunctionsofordersonΩisgivenby
BUCs(Ω;E):=u∈BUC[s](Ω;E):for|α|=[s]itholds[αu]BUCs−[s](Ω;E)<∞,
whereforτ∈(0,1)theseminorm[∙]BUCτ(Ω;E)isdefinedby
|u(x)−u(y)|
[u]BUCτ(Ω;E):=x,y∈Ωsup,x=y|x−y|τ,
andBUCs(Ω;E)isequippedwiththenorm|u|BUC[s](Ω;E)+|α|≤[s][αu]BUCs−[s](Ω;E).For
k∈N0wefurtherdenoteby
Ck(Ω;E)andCk(Ω;E)
thespaceofk-timescontinuouslydifferentiablefunctionsonΩandΩ,respectively.For
s≥0thespaceofthelocallyHöldercontinuousfunctionsofordersonΩandΩare
denotedbyCs(Ω;E)andCs(Ω;E),respectively.IfΩisbounded,itholds
Cs(Ω;E)=BUCs(Ω;E),s≥0.
setrfurtheeWC∞(Ω;E):=Ck(Ω;E),Cc∞(Ω;E):=u∈C∞(Ω;E):suppu⊂Ω,
N∈k0

202

endixApp

andanalogouslyonedefinesC∞(Ω;E)andCc∞(Ω;E).
Forp∈[1,∞)theBanachspaceoftheE-valuedLp-spaceonΩisdefinedby
Lp(Ω;E):=u:Ω→Estronglymeasurable:|u|Lpp(Ω;E):=|u(x)|Epdx<∞,
Ωandisendowedwiththenorm|u|Lp(Ω;E).ThespaceL∞(Ω;E)isdefinedwiththeusual
modification.SinceEisassumedtobereflexive,forp∈(1,∞)thesespacesarealso
reflexive,withLp(Ω;E)∗=Lq(Ω;E∗)andp1+q1=1.ThesetCc∞(Ω;E)isdensein
Lp(Ω;E)forp∈[1,∞).Forthegeneraltheoryofvector-valuedLp-spaceswerefertothe
ChaptersIIIandIVof[32].
Fork∈N0andp∈[1,∞]theE-valuedSobolevspaceoverΩisdefinedby
Wpk(Ω;E):=u∈Lp(Ω;E):αuexistsweakly,αu∈Lp(Ω;E)for|α|≤k,
/p1andisendowedwiththenorm|u|Wpk(Ω;E):=|α|≤k|αu|Lpp(Ω;E),whichturnsitinto
space.hacBanaWefurtherdefinethefollowingE-valuedfunctionspaces:forp,q∈[1,∞)ands>0the
spacevBesoBsp,q(Ω;E):=Lp(Ω;E),Wp[s]+1(Ω;E)s,q,
]+1s[forp∈[1,∞)ands>0theBesselpotentialspace
Hps(Ω;E):=Lp(Ω;E),Wp[s]+1(Ω;E)s,
]+1s[andforp∈[1,∞)ands>0theSlobodetskiispace
Wps(Ω;E):=Wspk(Ω;E),s=k∈N,
Bp,p(Ω;E),s∈/N.
TheseBanachspacesformscalesaccordingtothegeneralpropertiesofinterpolationspaces
listedinAppendixA.2.SinceEisofclassHTitfurtherholdsthat
Wpk(Ω;E)=Hpk(Ω;E),k∈N,(A.4.1)
see[91,Satz3.6].UsuallytheBesovspacesoverRnaredefinedbyaLittlewood-Paley
decompositionandtheBesselpotentialspacesaredefinedusingtheFouriertransform[91,
Def.3.1],[75,Def.4.3],andthenthespacesoverdomainsaredefinedviarestriction[9,
Sec4].ButsinceEisassumedtobeofclassHTandweassumethat∂Ωissmooth,itis
equivalenttodefinethemviainterpolation,asin[45].Thiscanbeseenusing(A.4.1),the
characterizations[75,Thm.4.2,4.3/3]andtheinterpolationresults[75,Thm.4.3/2],[91,
3.21].SatzTheSlobodetskiispacesadmitfors∈/N0theintrinsicrepresentation
Wps(Ω;E)=u∈Wp[s](Ω,E):for|α|=[s]itholds[Dαu]Wps−[s](Ω;E)<∞,


A.4FunctionSpacesonDomainsandBoundaries

203

(A.4.2)

whereforτ∈(0,1)theseminorm[∙]Wpτ(Ω;E)isgivenby
p|u(x)−u(y)|Ep
[u]Wpτ(Ω;E):=ΩΩ|x−y|n+τpdxdy,(A.4.2)
andthespaceontheright-handsideaboveisequippedwiththenorm
|u|pWp[s](Ω;E)+[Dαu]pWps−[s](Ω;E)1/p,
|α|=[s]
cf.[8,Sec.1,5].AsinthescalarcasewehavetheSobolevembeddings[8,Eq.5.4]
nnWps(Ω;E)→Wqτ(Ω;E),s−p≥τ−q,s≥τ,p≥q,
furtherandnWps(Ω;E)→Cτ(Ω;E),s−p≥τ.
IfEisfinitedimensional,Ωisboundedandtheaboveinequalitiesarestrict,thenthese
embeddingsarecompactby[1,Thm.6.3]and[7,Sec.I.2.7].
Nowsupposethat∂Ωiscompact.Thenthereareafinitecollectionofcharts(Ui,ϕi)for
∂Ωwithcorrespondingparametrisationsgiandapartitionofunity{ψi}subordinateto
iUi.Forp∈[1,∞]thespacesLp(∂Ω;E)aredefinedinastandardwaywithrespectto
thesurfacemeasureon∂Ω.Moreover,fors>0andp,q∈[1,∞)wedefineasin[82,Def.
3.6.1]Bsp,q(∂Ω;E):=u∈Lp(∂Ω;E):(ψiu)◦gi∈Bsp,q(Rn−1;E)foralli,
|u|Bsp,q(∂Ω;E):=i|(ψiu)◦gi|Bsp,q(Rn−1;E),
Hps(∂Ω;E):=u∈Lp(∂Ω;E):(ψiu)◦gi∈Hps(Rn−1;E)foralli,
|u|Hps(∂Ω;E):=i|(ψiu)◦gi|Hps(Rn−1;E),
Cs(∂Ω;E):=u∈C(∂Ω;E):(ψiu)◦gi∈Cs(Rn−1;E)foralli,
|u|Cs(∂Ω;E):=i|(ψiu)◦gi|Cs(Rn−1;E),
whichareallBanachspaceswiththeirrespectivenorms.Notethathereweidentifythe
functions(ψiu)◦giwiththeirtrivialextensiontoRn−1.Ifonechoosesanothercollectionof
chartsandanotherpartitionofunityfor∂Ω,oneobtainsthesamespaceswithequivalent
.elyectivrespnorms,Itfollowsfromthedefinitionsthatthebasicembeddingsobtainedfrominterpolationas
wellastheSobolevembeddingsforspacesoverdomainscarryovertothecorresponding
spacesoveraboundary,withdimensionnreplacedbyn−1.Moreover,asin[86,Thm.
4.3]itcanbeseenthat
C∞(∂Ω;E)→dBsp,q(∂Ω;E),Hps(∂Ω;E),
fors>0andp,q∈[1,∞).
Weconsiderlocalpropertiesoftheabovefunctionspacesonandneartheboundary.

204

endixApp

LemmaA.4.1.Lets>0andp,q∈[1,∞),andlet(U,φ)beachartfor∂Ω,with
correspondingpushforwardoperatorΦ.ThenΦinducesacontinuousisomorphism
Bsp,q(Ω∩U;E)→Bsp,q(R+n∩ϕ(U);E),
withinverseΦ−1.Moreover,forφ∈Cc∞(U)themapu→Φ(φu)iscontinuous
Bsp,q(∂Ω;E)→Bsp,q(Rn−1∩ϕ(U);E),
andforφ∈Cc∞(Rn−1∩ϕ(U))themapu→Φ−1(φu)iscontinuous
Bsp,q(Rn−1∩ϕ(U);E)→Bsp,q(∂Ω;E).
AlltheseassertionsremaintrueifonereplacestheBsp,q-spacesbytheHsp-spaces,s≥0.
Proof.Inthescalar-valuedcaseandforp=q=2,firstassertionisshownin[86,Thm.
4.1].TheproofforWps-spaceswiths∈N0andp∈(1,∞)carriesovertothevector-valued
case,fromwhichthegeneralcasefollowsfrominterpolation.Theremainingtwoassertions
followimmediatelyfromthedefinitionsofthespacesover∂Ω.
WefinishthissectionwithageneralinterpolationresultfortheH-andtheB-spaces.
PropositionA.4.2.LetEbeofclassHT,letΩ⊂Rnbeadomainwithsmoothboundary,
andletp∈[1,∞),0≤s1<s2,θ∈(0,1)ands=(1−θ)s1+θs2.Thenitholds
Hps1(Ω;E),Hps2(Ω;E)θ=Hps(Ω;E),Hps1(Ω;E),Hps2(Ω;E)θ,p=Bsp,p(Ω;E),
Bsp,p1(Ω;E),Bsp,p2(Ω;E)θ,p=Bsp,p(Ω;E),Bsp,p1(Ω;E),Bsp,p2(Ω;E)θ=Bsp,p(Ω;E),
wherethecases1=0isexcludedfortheB-spaces.Theseidentitiesremaintrueifone
replacesΩ=Rnbyitsboundary∂Ω.
Proof.(I)FirstletΩ=Rn.ThecomplexinterpolationresultfortheH-spacesisshown
in[91,Satz3.21].FortherealinterpolationoftheH-spacesweconsidertherealizationof
theshiftedLaplacianA:=1−ΔnonLp(Rn;E)withdomainD(A)=Wp2(Rn;E),which
isinvertibleandadmitsaboundedH∞-calculusofH∞-angleequaltozero,dueto[24,
Theorem5.5].ItthusfollowsfromcomplexinterpolationoftheH-spacesand(A.3.1)that
D(Aτ/2)=Hpτ,τ≥0.Fromthereiterationtheoremweinfer
Hps1,Hps2θ,p=D(As1/2),D(As2/2)θ,p=Lp,D(A([s2]+1)/2)s,p=Bsp,p,
]+1s[2asasserted.TherealinterpolationresultfortheB-spacesisshownin[75,Thm.4.2].Taking
powersofAasisomorphismsweobtainfrom[24,Prop.2.11]andtheinterpolationresults
thatwerealreadyshownthattherealizationofAτ/2onBτp,p1withdomainBτp,p1+τadmits
aboundedH∞-calculusofH∞-angleequaltozeroaswell,τ,τ1>0.Thusthecomplex
interpolationresultfollowsfromYagi’stheorem(A.3.1).
(II)NowsupposethatΩisadomainwithsmoothboundary.Forgivenk∈Nthereis
acontinuousextensionoperatorEΩfromWpl(Ω;E)toWpl(Rn;E)foralll∈{0,...,k},
whichmaybeextendedtotheH-andtheB-scalebyinterpolation.Italsofollowsfrom

A.5DifferentialOperatorsonaBoundary

205

interpolationthattherestrictionoffunctionsonRntoΩiscontinuousonbothscales.
FromtheresultonRnwethusinferthatEΩiscontinuous
Hps1(Ω;E),Hps2(Ω;E)θ→Hps(Rn;E),
andcombinedwiththerestrictiontoΩthisyieldsthatHps1(Ω;E),Hps2(Ω;E)θembeds
continuouslyintoHps(Ω;E).Conversely,EΩmapscontinuously
Hps(Ω;E)→Hps1(Rn;E),Hps2(Rn;E)θ,
andtherestrictionmapsthelatterspaceintoHps1(Ω;E),Hps2(Ω;E)θ.Wethusobtain
theassertedcomplexinterpolationresultfortheH-spaces.Theremainingidentitiesfollow
ts.argumensamethefrom(III)Wefinallyconsiderthespacesover∂Ω.Describetheboundarybyafinitecollectionof
charts(Ui,ϕi)withcorrespondingpush-forwardoperatorsΦi,andlet{ψi}beapartition
ofunitysubordinatetoiUi.Choosefurtherφi∈Cc∞(Rn−1∩ϕ(Ui))withφi≡1on
iΦi−1φiΦiψi.Foreachi,LemmaA.4.1showsthatthemapu→Φiψiuiscontinuous
suppΦiψi∩Rn−1.WedecomposetheidentityonHpτ(∂Ω;E),τ≥0,accordingtoid=
Hpτ(∂Ω;E)→Hpτ(Rn−1∩ϕi(Ui);E)forallτ,andfrominterpolationandtheresulton
domainsweobtainthatitiscontinuous
Hps1(∂Ω;E),Hps2(∂Ω;E)θ→Hps(Rn−1∩ϕi(Ui);E).
LemmaA.4.1alsoyieldsthatu→Φ−1(φiu)iscontinuousHps(Rn−1∩ϕi(Ui);E)→
Hps(∂Ω;E),whichimpliesthatHps1(∂iΩ;E),Hps2(∂Ω;E)θ→Hps(∂Ω;E).Theconverse
embeddingandtheremainingidentitiesareshowninthesameway.

A.5DifferentialOperatorsonaBoundary
LetΩ⊂Rnbeadomainwithcompactsmoothboundary∂Ω,andletEbeaBanachspace
ofclassHT.Wecallalinearmap
C:C∞(∂Ω;E)→L1(∂Ω;E)
alineardifferentialoperatoron∂Ωoforderk∈N0,ifforalllocalcoordinatesgfor∂Ω
therearecoefficientscγg∈L1Rn−1∩ϕ(U);B(E),γ∈N0n−1,|γ|≤ksuchthat
Cu◦g=cγg(x)nγ−1(u◦g)(x),x∈g−1(U∩∂Ω),(A.5.1)
k|≤γ|forallu∈C∞(∂Ω;E).Heren−1=(∂1,...,∂n−1)istheeuclidiangradientonRn−1.
Ofcourseitisunderstoodthatatleastonetopordercoefficientisnontrivial.Thelocal
coefficientscγgmaydependonthechosencoordinatesg.WedonotassumethatChas
globalcoefficients,inthesensethattherearecγ∈L1Γ;B(E)withcγg=cγ◦g.

206

endixApp

TheregularityofthelocalcoefficientsdetermineswhetherCcanbecontinuouslyextended
tootherfunctionspaces,oreventoawholescale.Supposethatforallcoordinatesgand
allφ∈Cc∞(U)thereisanestimateoftheform
|(φCu)◦g|Wps(Rn−1∩ϕ(U);E)|u|Wps+k(∂Ω;E),u∈C∞(∂Ω;E),
wheres≥0.ThenbydensityofC∞(∂Ω;E)inWps+k(∂Ω;E)theoperatorCmaybe
uniquelyextendedtoaboundedlinearmap
Wps+k(∂Ω;E)→Wps(∂Ω;E).
Inlocalcoordinatestheextendedoperatorisofthesameformasin(A.5.1)forsmooth
functions,asadensityargumentshows.Thisreasoningremainsvalidfortheextensionof
CtoBesovandBesselpotentialspaces.
AsufficientconditionfortheextendabilityofCtoWps+k(∂Ω;E)isthatforalll∈{0,...,k}
andallcoordinatesgthecoefficientscγgwith|γ|=larepointwisemultipliersfromWps+l
toWps.Inparticular,ifthelocalcoordinatesaresmooththenCextendstothewholescale
ofSlododetskii,BesovandBesselpotentialspaces,respectively.
Weconsiderexamplesfordifferentialoperatorsonboundaries.Forx∈∂Ωascalar-valued
functionu∈C∞(∂Ω)inducesanelementofthedualspaceofTx∂Ωviathedirectional
derivativeoftangentialvectorsatx∈∂Ω.ConsideringTx∂ΩasaHilbertspacewiththe
scalarproductinducedfromRn,thesurfacegradientΓu(x)ofuatxisthentheunique
elementofTx∂ΩcorrespondingtothisdualspaceelementviatheRieszisomorphism.In
localcoordinatesgfor∂Ω,withfundamentalformG=(gij)andinverseG−1=(gij),the
componentsofthesurfacegradientwithrespecttothecanonicalbasis{∂1g,...,∂n−1g}of
Tx∂ΩaregivenbythecomponentsofG−1n−1(u◦g)T,i.e.,
1−n∂Ωu◦g=gij∂j(u◦g)∂ig.(A.5.2)
=1i,jNowletEbeaBanachspace.Wedefinethesurfacegradientforafunctionu∈C∞(∂Ω;E)
incoordinatesbytheformula(A.5.2),whichyieldsthat
∂Ωu∈C∞(∂Ω;En).
TheapplicationoffunctionalsandtheHahn-Banachtheoremshowthatthisdefinitionis
independentofthechosencoordinates,sinceitisindependentoftheminthescalarvalued
case.Moreover,forα∗∈E∗andu∈C∞(∂Ω;E)itholds
α∗∂Ωu=∂Ωα∗u,
whereontheleft-handsidethefunctionalisappliedcomponentwisetoelementsofEn.In
thissensethedefinitionofthesurfacegradientforE-valuedfunctionsisconsistentwith
thedefinitioninthescalarcase,andbecauseofthiswestillspeakofagradientalthough
aRieszisomorphismisonlyindirectlyinvolved.

Inequalitiesergrdo-NirenbGagliaA.6

207

Foramultiindexγ∈N0n−1theoperator∂γΩisdefinedbytakingiterativelythecomponents
of∂Ωu.ThisyieldsalinearmapfromC∞(∂Ω;E)intoitself,andisthusaboundary
differentialoperatorintheabovesense.Inparticular∂γΩextendstoaboundedlinear
mapWps+|γ|(∂Ω;E)→Wps(∂Ω;E),s≥0,p∈[1,∞],
andanalogouslyfortheBesovandBesselpotentialscale.
Thesurfacedivergencediv∂Ωvofatangentialvectorfieldv∈C∞(∂Ω,Rn),i.e.,v(x)∈
Tx∂Ωforx∈∂Ω,isincoordinatesggivenby
1−ndiv∂Ωv◦g=1∂i|G|vi◦g,
G||=1iwhereviarethecomponentsofvwithrespecttothebasis{∂1g,...,∂n−1g}ofTx∂Ω.Itcan
beshownthediv∂Ωisindependentofthecoordinates.TheLaplace-Beltramioperator
Δ∂Ω:=div∂Ω∂Ω
isthenforu∈C∞(∂Ω)inlocalcoordinatesoftheform
1−nΔ∂Ωu◦g=1∂i|G|gij∂j(u◦g).
|G|i,j=1
TheLaplace-Beltramioperatorofavector-valuedfunctionu∈C∞(∂Ω;E)isdefinedin
coordinatesbytheaboveformula.Inthesamewayasforthesurfacegradientweseethat
thisdefinitionisindependentofthecoordinates,anditholds
α∗Δ∂Ωu=Δ∂Ωα∗u,α∗∈E∗,
whichshowsconsistencytothescalar-valuedcaseasabove.
WiththeLaplace-Beltramioperatorandthesurfacegradientonecandefinetheboundary
analogontogeneral’ellipticdifferentialoperators’actingonvector-valuedfunctions.We
finallyremarkthattheconsiderationsofthissectioncarryovertoageneralRiemannian
manifold.

InequalitiesergbGagliardo-NirenA.6ThefirstversionoftheGagliardo-Nirenberginequalityforintegerdifferentiabilitiesistaken
10.1].Thm.[41,fromPropositionA.6.1.LetΩ⊂Rnbeaboundeddomainwithsmoothdomain∂Ω,andlet
theintegersk∈N0,m∈N,withk<m,andthenumbersp,q,r∈[1,∞]satisfy
k−n=θm−n−(1−θ)n,
rqp

208

endixApp

whereθ∈[k/m,1]ifm−k−n/r∈/N0,andθ=k/mifm−k−n/r∈N0.Thenthereis
aconstantC>0suchthat
|u|Wpk(Ω)≤C|u|θWqm(Ω)|u|L1r−θ(Ω)forallu∈Wqm(Ω)∩Lr(Ω).
In[2,Prop.4.1]anpartialextensiontofractionalorderSlobodetskiispacesisgiven.Ob-
servethatforintegerdifferentiabilitiesPropositionA.6.1mayleadtoastrongerresult,for
instanceforp=q=2,r=1,k=0,m=1andθ=nn+2.
PropositionA.6.2.LetΩ⊂Rnbeadomainwithcompactsmoothboundary∂Ω,and
letthenumbersθ∈[0,1],s,s0,s1≥0withs0=s1andp,p1∈(1,∞),p0∈[1,∞),satisfy
1≤θ+1−θ,(A.6.1)
ppp01andnnn
s−p<θs1−p1+(1−θ)s0−p0,(A.6.2)
withthefollowingexceptions:itholdss0=0ifp0=1,itholdsθ>0ifs0=0,andit
holdsθ<1ifs1=0.ThenthereisaconstantC>0suchthat
|u|Wps(Ω)≤C|u|θWps11(Ω)|u|1W−ps0θ0(Ω)forallu∈Wps11(Ω)∩Wps00(Ω).
IfΩisboundedands<θs1+(1−θ)s0then(A.6.1)isnotnecessary.Further,theequality
signin(A.6.2)ispermittedifp0>1andeithers0,s1∈Norθs1+(1−θ)s0∈/N.
Bydefinitiontheseinequalitiescarryovertothespacesover∂Ω,withnreplacedbyn−1,
.elyectivresp

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