Regularity results for minimizers of integrals with (2,q)-growth in the Heisenberg group [Elektronische Ressource] / vorgelegt von Anna Föglein

Regularity results for minimizersof integrals with.2; q/-growth in the Heisenberg groupDer Naturwissenschaftlichen Fakultat¨der Friedrich-Alexander-Universitat¨ Erlangen-Nurnber¨ gzurErlangung des Doktorgradesvorgelegt vonAnna Foglein¨aus BudapestAls Dissertation genehmigt von derNaturwissenschaftlichen Fakultat¨ der Universitat¨ Erlangen-Nurnber¨ gTag der mundlichen¨ Prufung:¨ 18. Februar 2009Vorsitzender der Promotionskommission: Prof. Dr. Eberhard Bansch¨Erstberichterstatter: Prof. Dr. Frank DuzaarZweitberichterstatter: Prof. Dr. Andreas GastelZusammenfassungIn der vorliegenden Arbeit beschaftigen¨ wir uns mit einer Fragestellung aus der Regularitats-¨theorie fur¨ Minimierer von anisotrop wachsenden autonomen Integralfunktionalen. Die zulas-¨sigen Funktionen sind dabei auf der Heisenberg-Gruppe definiert.n 2nC1Fur¨ n2 N bezeichne H ’ R die n-dimensionale Heisenberg-Gruppe; weiter sei XD.X ;:::; X / der horizontale Gradient, und TD@ die vertikale Ableitung. Wir betrachten1 2n 2nC1Funktionale der Form ZF.u; / D f.Xu/ dx;definiert auf den schwach horizontal differenzierbaren Funktionen u : ! R auf einemn 2 2nbeschrankten¨ Gebiet H . Der C -Integrand f : R ! R erfulle¨ dabei fur¨ ein Paarvon Vorfaktoren 0< 1 L und einen Exponenten q > 2 die.2; q/-Wachstumsbedingungq2 2 2 2njzj f.z/ L 1Cjzj 8 z2Rund die entsprechende Konvexitatsbedingung¨q2 2 2 2 2n2jj D f.
Publié le : jeudi 1 janvier 2009
Lecture(s) : 15
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Source : D-NB.INFO/993665411/34
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Regularity results for minimizers
of integrals with.2; q/-growth in the Heisenberg group
Der Naturwissenschaftlichen Fakultat¨
der Friedrich-Alexander-Universitat¨ Erlangen-Nurnber¨ g
zur
Erlangung des Doktorgrades
vorgelegt von
Anna Foglein¨
aus BudapestAls Dissertation genehmigt von der
Naturwissenschaftlichen Fakultat¨ der Universitat¨ Erlangen-Nurnber¨ g
Tag der mundlichen¨ Prufung:¨ 18. Februar 2009
Vorsitzender der Promotionskommission: Prof. Dr. Eberhard Bansch¨
Erstberichterstatter: Prof. Dr. Frank Duzaar
Zweitberichterstatter: Prof. Dr. Andreas GastelIn der vorliegenden Arbeit beschaftigen¨ wir uns mit einer Fragestellung aus der Regularitats-¨
theorie fur¨ Minimierer von anisotrop wachsenden autonomen Integralfunktionalen. Die zulas-¨
sigen Funktionen sind dabei auf der Heisenberg-Gruppe definiert.
n 2nC1Fur¨ n2 N bezeichne H ’ R die n-dimensionale Heisenberg-Gruppe; weiter sei XD
.X ;:::; X / der horizontale Gradient, und TD@ die vertikale Ableitung. Wir betrachten1 2n 2nC1
Funktionale der Form Z
F.u; / D f.Xu/ dx;

definiert auf den schwach horizontal differenzierbaren Funktionen u : ! R auf einem
n 2 2nbeschrankten¨ Gebiet H . Der C -Integrand f : R ! R erfulle¨ dabei fur¨ ein Paar
von Vorfaktoren 0< 1 L und einen Exponenten q > 2 die.2; q/-Wachstumsbedingung
q
2 2 2 2n
jzj f.z/ L 1Cjzj 8 z2R
und die entsprechende Konvexitatsbedingung¨
q
2 2 2 2 2n2
jj D f.z/ L 1Cjzj jj 8 z;2R :
Das Hauptresultat dieser Arbeit ist folgender Regularitatssatz¨ fur¨ Minimierer vonF:
1;2Satz Sei u2 H W . / lokaler Minimierer vonF unter obigen Bedingungen, und der obere
Wachstumsexponent q erfulle¨
n o2nC 1 8
2< q < min 2C ; 2C :
.2n 1/.nC 1/ 9
Dann ist die volle Ableitung Du des Minimierers lokal Holder¨ -stetig. Dabei sind die horizontale
Ableitung Xu und die vertikale Ableitung T u lokal beschrankt,¨ und konnen¨ durch F.u/ und
kuk 2 abgeschatzt¨ werden. Zusatzlic¨ h ist Du schwach horizontal differenzierbar mit AbleitungL
2lokal in L , und ist analog abzuschatzen.¨
Fur¨ Minimierer von Funktionalen mit Standard-p-Wachstum in der Heisenberg-Gruppe mit
Wachstumsexponent 2 p < 4 wurde die Holderstetigk¨ eit der vollen Ableitung durch Min-
ngione, Zatorska und Zhong [52] gezeigt. Aus der Analysis auf dem EuklidischenR ist bekannt,
dass die Regularitat¨ von Minimierern von Funktionalen mit.p; q/-Wachstum dann gesichert ist,
wenn der Abstand zwischen p und q unter bestimmten Schranken bleibt (siehe z.B. [17]). Der
hier gezeigte Satz kombiniert diese beiden Situationen, und gibt eine Schranke an den oberen
iii
Zusammenfassungiv
Wachstumsexponenten q an, welche sicherstellt, dass Minimierer vonF Holder¨ -stetige volle
Ableitung besitzen.
Fur¨ den Beweis des Satzes werden Techniken aus der Regularitatstheorie¨ auf der Heisenberg-
Gruppe mit solchen aus der Analyse anisotroper Funktionale kombiniert: Wir approximieren
¨zunachst das eigentliche Funktional F durch regularisierte Funktionale mit Standardwachs-
tum, um schon bekannte Integrabilitats-¨ und Regularitatsresultate¨ fur¨ deren Minimierer ein-
setzen zu konnen.¨ In einer Folge von Interpolationsargumenten und abwechselndem Betra-
chten der vertikalen und horizontalen Ableitungsrichtungen werden dann schrittweise Regu-
laritatseigenschaften¨ der approximierenden Minimierer nur unter Verwendung der ursprung-¨
lichen.2; q/-Wachstumsbedingungen gewonnen, wobei fur¨ einige dieser Schritte Bedingungen
an q notwendig werden. Zuletzt wird der Minimierer vonF durch eine Folge von regularisierten
Minimierern approximiert, wobei die fur¨ letztere gezeigten Regularitatseigenschaften¨ erhalten
bleiben.
Ein Nebenresultat der Arbeit ist die lokale Beschranktheit¨ des Minimierers unter wesentlich
schwacheren¨ Bedingungen an das FunktionalF; diese wird unabhangig¨ von den Regularisierun-
gen fur¨ die Verwendung im Approximationsschritt gezeigt.
Satz Es sei 1< p< q, und der konvexe Integrand f sei erfulle¨ die.p; q/-Wachstumsbedingung
p q
2 2 2n2 2
1Cjzj f.z/ L 1Cjzj 8 z2R :
Dann gibt es eine Schranke p .n/> p, so dass gilt: Ist q < p , dann sind Minimierer vonF
lokal beschrankt.¨In the present thesis we are concerned with the regularity theory for minimizers of autonomous
integral functionals with anisotropic growth, where the admissible functions are defined on the
Heisenberg group.
n 2nC1For given n2 N, we denote by H ’ R the n-dimensional Heisenberg group, by XD
.X ;:::; X / the horizontal gradient, and by T D @ the vertical derivative. We consider1 2n 2nC1
functionals of the form Z
F.u; / D f.Xu/ dx

acting on horizontally weakly differentiable functions u : ! R on a bounded open domain
n 2 2n
H . The C -integrand f :R !R is assumed to satisfy the.2; q/-growth condition
q
2 2 2n2
jzj f.z/ L 1Cjzj 8 z2R
and the matching convexity condition
q
2 2 2 2 2n2
jj D f.z/ L 1Cjzj jj 8 z;2R :
for some exponent q > 2 and parameters 0< 1 L.
The main result of this thesis is the following regularity theorem for minimizers ofF:
1;2Theorem Let u2 H W . / be a local minimizer of the functional F fullfilling the above
growth and convexity conditions, and suppose that the upper growth exponent q satisfies
n o
2nC 1 8
2< q < min 2C ; 2C :
.2n 1/.nC 1/ 9
Then the full gradient Du is locally Holder¨ continuous on, and the horizontal gradient Xu
and the vertical derivative T u are bounded and can be estimated in terms ofF.u/ andkuk 2.L
2Moreover, Du is horizontally weakly differentiable with derivative in L and can be estimated
analogously.
For minimizers of functionals with standard p-growth in the Heisenberg group, Mingione,
Zatorska and Zhong [52] proved that the full gradient is Holder¨ continuous, provided that
n2 p< 4. In the setting of EulclideanR it is known that the regularity of minimizers of func-
tionals with.p; q/-growth hinges on the smallness of the gap between p and q (see e.g. [17]).
The above theorem combines these two situations, and gives a bound on q which ensures that
minimizers ofF possess Holder¨ continuous full derivative.
v
Abstractvi
The proof of the theorem merges techniques from regularity theory in the Heisenberg group
setting with those typical for the analysis of anisotropic functionals: We start by approximating
the original functional by regularized standard-growth versions in order to employ the already
known regularity theory for their minimizers in a qualitative way. For these more regular func-
tionals, we recover the properties that are preserved if one only assumes the initial.2; q/-growth
conditions. This is done in a series of interpolation arguments that improve the regularity of the
vertical derivative and the horizontal gradient in turn; some of these steps require the introduc-
tion of bounds on q. Finally, the original minimizer is approximated by regularized minimizers
in a way that preserves their regularity that we have proved for the latter when passing to the
limit.
A side result of the present thesis is the local boundedness of the minimizer under considerably
weaker conditions on the functionalF than those needed for the main theorem. This bounded-
ness is used in the approximation procedure, and we prove it independently of the regulariza-
tions.
Theorem Let 1 < p < q, and let the integrand f be convex and satisfy the .p; q/-growth
condition
p q
2 2 2 2 2n
1Cjzj f.z/ L 1Cjzj 8 z2R :
There exists a bound p .n/ > p such that there holds: If q < p , then minimizers ofF are
locally bounded.1.1. Statement of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2. Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1. Properties of the Heisenberg groups . . . . . . . . . . . . . . . . . . . . . . . 11
2.2. Horizontal Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3. Regularity results for q-growth equations . . . . . . . . . . . . . . . . . . . . 16
2.4. Technical and Iteration Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1. Definition of the regularized functionalsF . . . . . . . . . . . . . . . . . . . 25

24.2. A -independent local L estimate for T u . . . . . . . . . . . . . . . . . . . . 27
4.3. Testing the equation with derivatives . . . . . . . . . . . . . . . . . . . . . . . 36
4.4. Basic higher integrability of the gradient . . . . . . . . . . . . . . . . . . . . . 39
4.5. Local boundedness of the vertical derivative T u . . . . . . . . . . . . . . . . . 57
4.6. Local of the horizontal gradient Xu . . . . . . . . . . . . . . . . 61
5.1. Mollification in the Heisenberg group . . . . . . . . . . . . . . . . . . . . . . 77
5.2. Approximation procedure (Proof of Lemma 5.1) . . . . . . . . . . . . . . . . . 81
6.1. Construction of a regularized integrand . . . . . . . . . . . . . . . . . . . . . . 87
6.2. Application of results concerning q-growth problems . . . . . . . . . . . . . . 90
A.1. Supplementary calculations for Lemma 4.3 . . . . . . . . . . . . . . . . . . . 91
A.2. Supplementary Calculations for Corollary 4.5 . . . . . . . . . . . . . . . . . . 95
A.3. calculations for 4.7 . . . . . . . . . . . . . . . . . . 96
vii
continuittheContentsofximationsLoBoundednesgradient87gulafo1yHrittheRegulaApp4.y19functionalsminimizerIntro11Prelimina6.Vitaecal99older3.yof25fullfunctionals5.rizedroregulabofrersrizede77z1.iductionm2.minirriesBibliographyA.CurriculumApp103endix91A number of natural phenomena are modeled by integral functionals and partial differential
equations. We quantify the laws that govern the physical world, the interplay of forces and en-
ergy, with the help of these mathematical tools. A given configuration is usually described by a
function u, which assigns certain properties to each point in time and space. Physical character-
istics of the situation, such as its inherent energy or forces acting, are then calculated in terms
of u and its derivatives. Generally, stable configurations of a system are assumed to correspond
to minimum points of the functional associated with its energy; and what one intuitively sees as
the equilibrium point of some “continuous” situation corresponds to a solution of the equation
that balances the forces involved. Similar approaches can be applied to other subjects possess-
ing more or less continuous dynamics, from economic studies to biological models to military
tactics. Due to this wide applicability, the Calculus of Variations and differential equations have
been extensively studied since the beginnings of differential calculus.
Let us first consider a typical first-order integral functional:
Z
F.u; / D f.x; u; Du/ dx;

nwhere u :R !R is a differentiable function on the domain describing some configu-
nration, and f :RR !R gives the energy density in each point, depending on u and its
derivative Du.
For a given functionalF, two fundamental questions arise: Is there a minimizer at all? And if
yes, how regular is it depending on the properties of f ? These two questions are more deeply
linked than is visible at first glance.
Before we can set out to investigate properties of a minimizer, it has to be ensured that one exists
at all. In most practical situations, it is not possible to construct a minimizer explicitly to prove
that it exists; instead, methods from functional analysis are invoked. These methods tend to rely
on finding a sequence of functions that converges to the desired solution or minimizer in some
sense. Clearly, this approach requires that the approximating functions as well as the minimizer
lie in an appropriately complete space. Unfortunately, the most natural spaces to consider for
ksolutions of differential equations, that is to say the spaces C of k times differentiable func-
tions, are not suitably complete. This difficulty is bypassed by the introduction of the Sobolev
k;pspaces of “weakly differentiable” functions W . While the completeness of these spaces now
guarantees the existence of a generalized “weak” solution, we pay the price when it comes to
the second fundamental question: A priori, Sobolev functions are not even continuous, much
less differentiable. Therefore, as an important part of the general question of regularity, one has
to ask in what sense a weak solution is a solution of the initial problem at all, and under what
conditions a weak solution is regular enough to be a classical solution.
1
duIntroction1.2 1. Introduction
For an in-depth overview of the development of regularity theory in various settings, explaining
the underlying mechanisms and connections, we refer to the survey article [50]. Here, we shall
concentrate on those aspects that directly relate and lead up to the topic of the present work.
A well-understood standard model case in many physical applications is represented by au-
tonomous functionals with p-growth,
Z
F.u; / D f.Du.x// dx;

n 2 nwhere only the derivative Du of u :R !R plays a role, and the C -integrand f :R !R is
uniformly convex with p-growth, i.e., there exist parameters 0< 1 L and p> 1 with
p
p 2 2
jzj f.z/ L 1Cjzj ;
and
p 2
p 2 2 2 2 22
jzj jj hD f.z/ ;i L 1Cjzj jj :
1;pMinimizers of such functionals with respect to W boundary conditions are known to be
Holder¨ continuous. In fact, we could even allow more general integrands f.x; u; Du/ that are
continuous in.x; u/, and still obtain Holder¨ continuity [40]; it is also possible to relax the regu-
larity requirements on f in the Du variable without losing too much regularity of the minimizer
(for details, see [50]). These possible generalizations suggest that the essential prerequisite
for regularity is the growth of f with respect to the Du variable. The growth conditions are
exploited in two ways: In the first step, the p-growth from below, combined with convexity
1;p
guarantees the existence of a minimizer uN in W . In a second step, one can combine the upperloc
2and lower bounds on f and D f to prove the regularity of uN. A technically important role is
played by the fact that the weak derivative DuN possesses enough integrability to be substituted
for z in the growth conditions, and also that the homogeneities of the upper and lower bounds
1;pmatch. In the final estimate, the Holder¨ semi-norm of uN is controlled in terms of the W -norm
of uN, including some constants that depend on the growth exponent p, the dimension n and also
on the ellipticity ratio L= .
While the class of p-growth integrands is suitable to describe a number of situations, in elas-
ticity theory one frequently encounters anisotropic integrands f whose upper and lower growth
exponents do not match. Considering the associated.p; q/-growth conditions
q
p 2 2
jzj f.z/ L 1Cjzj ;
and
q 2
p 2 2 2 2 22
jzj jj hD f.z/ ;i L 1Cjzj jj
(p,q)-growthintegrands

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