Line–energy Ginzburg–Landau models: zero–energy states

Pierre-Emmanuel Jabin

Description

Niveau: Supérieur, Doctorat, Bac+8
Line–energy Ginzburg–Landau models: zero–energy states Pierre-Emmanuel Jabin*, email: Felix Otto** email: Benoıt Perthame* email: *Departement de Mathematiques et Applications, Ecole Normale Superieure 45 rue d'Ulm, 75230 Paris Cedex 05, France **Institut fur Angewandte Mathematik, Universitat Bonn, Wegelerstrasse 10, 53115 Bonn, Germany Abstract We consider a class of two–dimensional Ginzburg–Landau prob- lems which are characterized by energy density concentrations on a one– dimensional set. In this paper, we investigate the states of vanishing energy. We classify these zero–energy states in the whole space: They are either con- stant or a vortex. A bounded domain can sustain a zero–energy state only if the domain is a disk and the state a vortex. Our proof is based on specific entropies which lead to a kinetic formulation, and on a careful analysis of the corresponding weak solutions by the method of characteristics. Key-words Ginzburg–Landau energy, vortices, kinetic formulation, ferro- magnetism. AMS Cl. 35B65, 35J60, 35L65, 74G65, 82D30. 1

• then either


• trivially extended


• ginzburg–landau models


• either con- stant


• consider any


• connected domain

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 8
• Langue: English

Extrait

Line–energyGinzburg–Landau models:zero–energystates Pierre-EmmanuelJabin*, email:[e-mail] FelixOtto** email:[e-mail] BenoˆtPerthame* email:[e-mail] *DepartementdeMathematiquesetApplications,  EcoleNormaleSuperieure 45rued’Ulm,75230ParisCedex05,France **InstitutfurAngewandteMathematik,UniversitatBonn, Wegelerstrasse10,53115Bonn,Germany Abstract Weconsideraclassoftwo–dimensionalGinzburg–Landauprob- lemswhicharecharacterizedbyenergydensityconcentrationsonaone– dimensionalset.Inthispaper,weinvestigatethestatesofvanishingenergy. Weclassifythesezero–energystatesinthewholespace:Theyareeithercon- stantoravortex.Aboundeddomaincansustainazero–energystateonly ifthedomainisadiskandthestateavortex.Ourproofisbasedonspecic entropieswhichleadtoakineticformulation,andonacarefulanalysisof thecorrespondingweaksolutionsbythemethodofcharacteristics. Key-words Ginzburg–Landauenergy,vortices,kineticformulation,ferro- magnetism. AMSCl. 35B65,35J60,35L65,74G65,82D30. 1

1Introduction Line–energyGinzburg–Landaumodelsariseinmanyphysicalsituationslike smecticliquidcrystals,softferromagneticlms,inblisterformationor— moreabstractly—inthegradienttheoryofphasetransition(see[6]and thereferencestherein).Roughlyspeaking,thesemodelscomethroughdi- mensionalreductionofathreedimensionalGinzburg–Landau–typemodelin athinlmandsingularlydependonasmallparameterεproportionalto thelmthickness.Thesevariationalproblemshaveincommonthatinthe limitε↓0,theminimizersconvergetoatwo–dimensionalvectoreldofunit lengthwhichisdivergence–free.Vectoreldsofthisclassgenericallyhave linesingularities,whichtypicallyareimposedbytheboundaryconditions. Thisisre ectedinthephenomenonthatinthelimitε↓0,theenergydensity oftheminimizersconcentratesonaone–dimensionalset.Pointsingularities carryonlyavanishingfractionoftheenergy—asopposedtotheclassical Ginzburg–Landauproblem(seeF.Bethuel,H.BrezisandF.Helein[3]). 1.1Themodels Twoexamplesofline–energyGinzburg–Landaumodelshavebeenrecently considered. Model1 .(Jin&Kohn[10],Ambrosio,DeLellis&Mantegazza[1],DeS- imone,Kohn,Muller&Otto[5]).Theadmissibletwo–dimensionalvector eldsaregivenby divm=0in ,m
n=0on∂ ,(1.1) wherendenotestheouterunitnormaltotheboundary∂ of IR 2 ,and theenergyis 1ZZ E ε 1 (m)=ε|rm| 2 +(1�|m| 2 ) 2 . ε Model2 .(Riviere&Serfaty[11]).Theconstraintisgivenby |m|=1in , whereasthefunctionalis 1ZZ E ε 2 (m)=ε|rm| 2 +|r �1 divm| 2 , ε 2 RI 2 )2.1( )3.1( )4.1(