Convex Variational Problems: Linear, Nearly Linear and Anisotropic Growth Conditions

221 Pages·2003·1.085 MB·English

by  Michael Bildhauer (auth.)

1818
Lecture Notes in Mathematics
Editors:
J.--M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
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Michael Bildhauer
Convex Variational
Problems
Linear, Nearly Linear
and Anisotropic
Growth Conditions
1 3
Author
MichaelBildhauer
DepartmentofMathematics
SaarlandUniversity
P.O.Box151150
66041Saarbru¨cken
Germany
e-mail:[email protected]
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MathematicsSubjectClassification(2000):49-02,49N60,49N15,35-02,35J20,35J50
ISSN0075-8434
ISBN3-540-40298-5Springer-VerlagBerlinHeidelbergNewYork
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Preface
In recent years, two (at first glance) quite different fields of mathematical
interest have attracted my attention.
• Elliptic variational problems with linear growth conditions. Here the no-
tion of a “solution” is not obvious and, in fact, the point of view has to be
changed several times in order to get some deeper insight.
• The study of the smoothness properties of solutions to convex anisotropic
variational problems with superlinear growth.
It took some time to realize that, in spite of the fundamental differences and
with the help of some suitable theorems on the existence and uniqueness of
solutions in the case of linear growth conditions, a non-uniform ellipticity
condition serves as the main tool towards a unified view of the regularity
theory for both kinds of problems.
This is roughly speaking the background of my habilitations thesis at the
Saarland University which is the basis for this presentation.
Of course there is a long list of people who have contributed to this mono-
graph in one or the other way and I express my thanks to each of them.
Without trying to list them all, I really want to mention:
Prof. G. Mingione is one of the authors of the joint paper [BFM]. The valu-
able discussions on variational problems with non-standard growth conditions
go much beyond this publication.
Prof. G. Seregin took this part in the case of variational problems with
linear growth.
Large parts of the presented material are joint work with Prof. M. Fuchs:
this, in the best possible sense, requires no further comment. Moreover, I am
deeply grateful for the numerous discussions and the helpful suggestions.
Saarbru¨cken, April 2003 Michael Bildhauer

Dedicated to Christina

Contents
1 Introduction ............................................... 1
2 Variational problems with linear growth: the general setting 13
2.1 Construction of a solution for the dual problem which is of
class W1 (Ω;RnN) ..................................... 14
2,loc
2.1.1 The dual problem ................................. 14
2.1.2 Regularization .................................... 16
2.1.3 W1 -regularity for the dual problem ................ 19
2,loc
2.2 A uniqueness theorem for the dual problem ................. 20
2.3 Partial C1,α- and C0,α-regularity, respectively, for generalized
minimizers and for the dual solution ....................... 25
2.3.1 Partial C1,α-regularity of generalized minimizers ...... 26
2.3.2 Partial C0,α-regularity of the dual solution ........... 29
2.4 Degenerate variational problems with linear growth .......... 32
2.4.1 The duality relation for degenerate problems.......... 33
2.4.2 Application: an intrinsic regularity theory for σ ....... 39
3 Variational integrands with (s,μ,q)-growth ................ 41
3.1 Existence in Orlicz-Sobolev spaces ......................... 42
3.2 The notion of (s,μ,q)-growth – examples ................... 44
3.3 A priori gradient bounds and local C1,α-estimates for scalar
and structured vector-valued problems ..................... 50
3.3.1 Regularization .................................... 52
q
3.3.2 A priori L -estimates .............................. 54
3.3.3 Proof of Theorem 3.16 ............................. 61
3.3.4 Conclusion ....................................... 67
3.4 Partial regularity in the general vectorial setting............. 69
3.4.1 Regularization .................................... 69
3.4.2 A Caccioppoli-type inequality ....................... 70
3.4.3 Blow-up.......................................... 72
3.4.3.1 Blow-up and limit equation ................. 74
3.4.3.2 An auxiliary proposition .................... 76
3.4.3.3 Strong convergence ........................ 83
3.4.3.4 Conclusion................................ 86
3.4.4 Iteration ......................................... 87
X Contents
3.5 Comparison with some known results ...................... 89
3.5.1 The scalar case.................................... 89
3.5.2 The vectorial setting ............................... 90
3.6 Two-dimensional anisotropic variational problems ........... 91
4 Variational problems with linear growth: the case of
μ-elliptic integrands ....................................... 97
4.1 The case μ < 1+2/n ....................................100
4.1.1 Regularization ....................................101
4.1.2 Some remarks on the dual problem ..................101
4.1.3 Proof of Theorem 4.4 ..............................103
4.2 Bounded generalized solutions.............................104
4.2.1 Regularization ....................................108
4.2.2 The limit case μ = 3 ...............................111
4.2.2.1 Higher local integrability ...................111
4.2.2.2 The independent variable ...................113
p
4.2.3 L -estimates in the case μ < 3 ......................116
4.2.4 A priori gradient bounds ...........................118
4.3 Two-dimensional problems................................122
4.3.1 Higher local integrability in the limit case ............123
4.3.2 The case μ < 3....................................129
4.4 A counterexample .......................................132
5 Bounded solutions for convex variational problems with a
wide range of anisotropy...................................141
5.1 Vector-valued problems ..................................142
5.2 Scalar obstacle problems .................................149
6 Anisotropic linear/superlinear growth in the scalar case ...161
A Some remarks on relaxation ...............................173
A.1 The approach known from the minimal surface case..........174
A.2 The approach known from the theory of perfect plasticity.....176
A.3 Two uniqueness results...................................181
B Some density results.......................................185
B.1 Approximations in BV ...................................185
B.2 A density result for U ∩L(c) ..............................191
B.3 Local comparison functions ...............................194
C Brief comments on steady states of generalized Newtonian
fluids ......................................................199
D Notation and conventions..................................205
References.....................................................207
Index ..........................................................215