Convex variational problems

Lecture Notes in Mathematics Editors: J.--M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1818 3 Berlin Heid...

Author: Michael Bildhauer

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Lecture Notes in Mathematics Editors: J.--M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1818

3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo

Michael Bildhauer

Convex Variational Problems Linear, Nearly Linear and Anisotropic Growth Conditions

13

Author Michael Bildhauer Department of Mathematics Saarland University P.O. Box 151150 66041 Saarbrücken Germany e-mail: [email protected]

Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

Mathematics Subject Classification (2000): 49-02, 49N60, 49N15, 35-02, 35J20, 35J50 ISSN 0075-8434 ISBN 3-540-40298-5 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de c Springer-Verlag Berlin Heidelberg 2003 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10935590

41/3142/du - 543210 - Printed on acid-free paper

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 notion 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 monograph 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 valuable 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.

Saarbr¨ ucken, April 2003

Michael Bildhauer

Dedicated to Christina

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Variational problems with linear growth: the general setting 2.1 Construction of a solution for the dual problem which is of 1 (Ω; RnN ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . class W2,loc 2.1.1 The dual problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 -regularity for the dual problem . . . . . . . . . . . . . . . . 2.1.3 W2,loc 2.2 A uniqueness theorem for the dual problem . . . . . . . . . . . . . . . . . 2.3 Partial C 1,α - and C 0,α -regularity, respectively, for generalized minimizers and for the dual solution . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Partial C 1,α -regularity of generalized minimizers . . . . . . 2.3.2 Partial C 0,α -regularity of the dual solution . . . . . . . . . . . 2.4 Degenerate variational problems with linear growth . . . . . . . . . . 2.4.1 The duality relation for degenerate problems . . . . . . . . . . 2.4.2 Application: an intrinsic regularity theory for σ . . . . . . .

13

Variational integrands with (s, μ, q)-growth . . . . . . . . . . . . . . . . 3.1 Existence in Orlicz-Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The notion of (s, μ, q)-growth – examples . . . . . . . . . . . . . . . . . . . 3.3 A priori gradient bounds and local C 1,α -estimates for scalar and structured vector-valued problems . . . . . . . . . . . . . . . . . . . . . 3.3.1 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 A priori Lq -estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Proof of Theorem 3.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Partial regularity in the general vectorial setting . . . . . . . . . . . . . 3.4.1 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 A Caccioppoli-type inequality . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3.1 Blow-up and limit equation . . . . . . . . . . . . . . . . . 3.4.3.2 An auxiliary proposition . . . . . . . . . . . . . . . . . . . . 3.4.3.3 Strong convergence . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 42 44

3

14 14 16 19 20 25 26 29 32 33 39

50 52 54 61 67 69 69 70 72 74 76 83 86 87

X

Contents

3.5 Comparison with some known results . . . . . . . . . . . . . . . . . . . . . . 3.5.1 The scalar case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 The vectorial setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Two-dimensional anisotropic variational problems . . . . . . . . . . .

89 89 90 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 4.2.3 Lp -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

1 Introduction

One of the most fundamental problems arising in the calculus of variations is to minimize strictly convex energy functionals with respect to prescribed Dirichlet boundary data. Numerous applications for this type of variational problems are found, for instance, in mathematical physics or geometry. Here we do not want to give an introduction to this topic – we just refer to the monograph of Giaquinta and Hildebrandt ([GH]), where the reader will find in addition an intensive discussion of historical facts, examples and references. Let us start with a more precise formulation of the problem under consideration: given a bounded Lipschitz domain Ω ⊂ Rn , n ≥ 2, and a variational integrand f : RnN → R of class C 2 (RnN ) we consider the autonomous minimization problem f (∇w) dx −→ min (P) J[w] := Ω

among mappings w: Ω → RN , N ≥ 1, with prescribed Dirichlet boundary data u0 . Depending on f , the comparison functions are additionally assumed to be elements of a suitable energy class K. In the following, the variational integrand is always assumed to be strictly convex (in the sense of definition), thus we do not touch the quasiconvex case (compare, for instance, [Ev], [FH], [EG1], [AF1], [AF2], [CFM]). The purpose of our studies is to establish regularity results for (maybe generalized and not necessarily unique) minimizers of the problem (P) under linear, nearly linear and/or anisotropic growth conditions on f together with some appropriate notion of ellipticity: if u denotes a suitable (weak) solution of (P), then three different kinds of results are expected to be true. THEOREM 1 (Regularity in the scalar case) Assume that N = 1 and that f satisfies some appropriate growth and ellipticity conditions. Then u is of class C 1,α (Ω) for any 0 < α < 1.

M. Bildhauer: LNM 1818, pp. 1–12, 2003. c Springer-Verlag Berlin Heidelberg 2003

2

1 Introduction

According to an example of DeGiorgi (see [DG3], compare also [GiuM2], [Ne] and the recent example [SY]), there is no hope to prove an analogous result of this strength in the vectorial setting. Here we can only hope for THEOREM 2 (Partial regularity in the vector-valued case) Assume that N > 1 and that f satisfies some appropriate growth and ellipticity conditions. Then there is an open set Ω0 ⊂ Ω of full Lebesgue measure, i.e. |Ω − Ω0 | = 0, such that u ∈ C 1,α (Ω0 ; RN ), 0 < α < 1. Finally, an additional structure condition might improve Theorem 2 to full regularity (see [Uh], earlier ideas are due to [Ur]): THEOREM 3 (Full regularity in the vector-valued case with some additional structure) Suppose that in the vectorial setting the integrand f satisfies in addition f (Z) = g(|Z|2 ) for some function g: [0, ∞) → [0, ∞) of class C 2 (plus some H¨ older condition for the second derivatives). Then u is of class C 1,α (Ω; RN ), 0 < α < 1. As the essential assumptions, the growth and the ellipticity conditions on f are involved in the above theorems. Hence, in order to make our discussion more precise and to summarize the various cases for which Theorems 1–3 are known to be true, we first introduce some brief classification of the integrands under consideration with respect to both growth and ellipticity properties. We also remark that in the cases A and B considered below the existence (and the uniqueness) of minimizers in suitable energy spaces is easily established. Before going through the following list it should be emphasized that we do not claim to give an historical overview which is complete to some extent. A.1 Power growth Having the standard example fp (Z) = (1 + |Z|2 )p/2 , 1 < p, in mind, let us assume that the growth rates from above and below coincide, i.e. for some number p > 1 and with constants c1 , c2 , C, λ, Λ > 0 the integrand f satisfies for all Z, Y ∈ RnN (note that the second line of (1) implies the first one) c1 |Z|p − c2 ≤

f (Z)

≤ C 1 + |Z|p ,

p−2 p−2 λ 1 + |Z|2 2 |Y |2 ≤ D2 f (Z)(Y, Y ) ≤ Λ 1 + |Z|2 2 |Y |2 .

(1)

With the pioneering work of DeGiorgi, Moser, Nash as well as of Ladyzhenskaya and Ural’tseva, Theorem 1 is well known in this setting and of course many other authors could be mentioned (see [DG1], [Mos], [Na] and [LU1] for a complete overview and a detailed list of references). As already noted above, the third theorem in this setting should be mainly connected to the name of Uhlenbeck (see [Uh], where the full strength of (1) is not needed which means that also degenerate ellipticity can be considered).

1 Introduction

3

Without additional structure conditions in the vectorial case, the twodimensional case n = 2 substantially differs from the situation in higher dimensions: a classical result of Morrey ensures full regularity if n = 2 (here we like to refer to [Mor1], the first monograph on multiple integrals in the calculus of variations, where again detailed references can be found). Finally, Theorem 2 is proved in any dimension and in a quite general setting by Anzellotti/Giaquinta ([AG2]), where the whole scale of integrands up to the limit case of linear growth is covered (with some suitable notion of relaxation). In addition, the assumptions on the second derivatives are much weaker than stated above, i.e. their partial regularity result is true whenever D2 f (Z) > 0 holds for any matrix Z. To keep the historical line, we like to mention the earlier contributions on partial regularity [Mor2], [GiuM1], [Giu1] (compare also [DG2], [Alm], a detailed overview is found in [Gia1]). A.2 Anisotropic power growth The study of anisotropic variational problems was pushed by Marcellini ([Ma2]–[Ma7]) and is a natural extension of (1). To give some motivation we consider the case n = 2, 2 ≤ p ≤ q and replace fp by p q fp,q (Z) = 1 + |Z|2 2 + 1 + |Z2 |2 2 ,

Z = (Z1 , Z2 ) ∈ R2N ,

hence f is allowed to have different growth rates from above and from below. The natural generalization of the structure condition (1) is the requirement that f satisfies (again the growth conditions on the second derivatives imply the corresponding growth rates of f ) c1 |Z|p − c2 ≤

f (Z)

≤ C 1 + |Z|q ,

p−2 q−2 λ 1 + |Z|2 2 |Y |2 ≤ D2 f (Z)(Y, Y ) ≤ Λ 1 + |Z|2 2 |Y |2

(2)

for all Z, Y ∈ RnN , where, as usual, c1 , c2 , C, λ, Λ denote some positive constants and 1 < p ≤ q. If p and q differ too much, then it turns out that even in the scalar case singularities may occur (to mention only one famous example we refer to [Gia2]). However, following the work of Marcellini, suitable assumptions on p and q yield regular solutions (compare Section 3.5 for a discussion of these conditions). Note that [Ma5] also covers the case N > 1 with some additional structure condition. In the general vectorial setting only a few contributions are available, we like to refer to the papers of Acerbi/Fusco ([AF4]) and Passarelli Di Napoli/Siepe ([PS]), where partial regularity results are obtained under quite restrictive assumptions on p and q excluding any subquadratic growth (again see Section 3.5).

4

1 Introduction

If an additional boundedness condition is imposed, then the above results are improved by Esposito/Leonetti/Mingione ([ELM2]) and Choe ([Ch]). In [ELM2] higher integrability (up to a certain extent) is established (N ≥ 1, 2 ≤ p) under a quite weak relation between p and q. A theorem of the third type is found in [Ch]. B.1 Growth conditions involving N-functions Studying the monograph of Fuchs and Seregin ([FuS2]) it is obvious that many problems in mathematical physics are not within the reach of power growth models – the theories of Prandtl-Eyring fluids and of plastic materials with logarithmic hardening serve as typical examples. The variational integrands under consideration are now of nearly linear growth, for example we have to study the logarithmic integrand f (Z) = |Z| ln(1 + |Z|) which satisfies none of the conditions (1) or (2). The main results on integrands with logarithmic structure are proved by Frehse/Seregin ([FrS]: full regularity if n = 2), Fuchs/Seregin ([FuS1]: partial regularity if n ≤ 4), Esposito/Mingione ([EM2]: partial regularity in any dimension) and finally by Mingione/Siepe ([MS]: full regularity in any dimension). B.2 The first extension of the logarithm As a first natural extension one may think of integrands which are bounded from above and below by the same quantity A(|Z|), where A: [0, ∞) → [0, ∞) denotes some arbitrary N-function satisfying a Δ2 -condition (see [Ad] for the precise definitions). Although this does not imply some natural bounds (in terms of A) on the second derivatives, (1) and (2) suggest the following model: given a N-function A as above, positive constants c, C, λ and Λ, we assume that our integrand f satisfies c A(|Z|) ≤

μ 2 −2

λ 1 + |Z|

f (Z)

≤ C A(|Z|) ,

|Y | ≤ D f (Z)(Y, Y ) ≤ Λ 1 + |Z| 2

2

2

q−2 2

(3) |Y |

2

for all Z, Y ∈ RnN and for some real numbers 1 ≤ μ, 1 < q ≤ 2, this choice being adapted to the logarithmic integrand which satisfies (3) with μ = 1 and q = 1 + ε for any ε > 0. Note that the correspondence to (1) and (2) is only of formal nature: since we require μ ≥ 1, the μ-ellipticity condition, i.e. the first inequality in the second line of (3), does not give any information on the lower growth rate of f in terms of a power function with exponent p > 1. A first investigation of variational problems with the structure (3) under some additional balancing conditions is due to Fuchs and Osmolovskii ([FO]), where Theorem 2 is shown in the case that μ < 4/n.

1 Introduction

5

Theorems of type 1 and 3 are established by Fuchs and Mingione (see [FuM]) – their assumptions on μ and q are discussed in Section 3.5. C Linear growth It remains to discuss the case of variational problems with linear growth. On account of the lack of compactness in the non-reflexive Sobolev space W11 (Ω; RN ), the problem (P) in general fails to have solutions. Thus one either has to introduce a suitable notion of generalized minimizers (possibility i)) or one must pass to the dual variational problem (possibility ii)). ad i). Since the integrand f under consideration is of linear growth, any ◦

J-minimizing sequence {um }, um ∈ u0 + W11 (Ω; RN ), is uniformly bounded in the space BV (Ω; RN ). This ensures the existence of a subsequence (not relabeled) and a function u in BV (Ω; RN ) such that um → u in L1 (Ω; RN ). Thus, one suitable definition of a generalized minimizer u is to require u ∈ M, where the set M is given by M =

u ∈ BV Ω; RN : u is the L1 -limit of a J-minimizing sequence ◦ from u0 + W11 (Ω; RN ) .

Another point of view is to define a relaxed functional Jˆ on the space BV (Ω; RN ) (a precise notion of relaxation is given in Appendix A). Then generalized solutions of the problem (P) are introduced as minimizers of a ˆ relaxed problem (P). Remark 1.1. We already like to mention that these formally different points of view in fact lead to the same set of functions. Moreover, the third approach to the definition of generalized minimizers given in [Se1], [ST] also leads to the same class of minimizing objects. ad ii). Following [ET] we write J[w] =

sup

l(w, τ ) ,

τ ∈L∞ (Ω;RnN )

◦ w ∈ u0 + W11 Ω; RN ,

where l(w, τ ) denotes some natural Lagrangian (see Section 2.1.1). If we let R : L∞ Ω; RnN → R , R(τ ) :=

inf ◦

u∈u0 +W11 (Ω;RN )

then the dual problem reads as

l(u, τ ) =

⎧ ⎪ ⎪ ⎨ −∞ ,

if div τ = 0 ,

⎪ ⎪ ⎩ l(u0 , τ ) , if div τ = 0 ,

6

1 Introduction

to maximize R among all functions in L∞ Ω; RnN ,

(P ∗ )

where the existence of solutions easily is established. In any of the above definitions the set of generalized minimizers of the problem (P) may be very “large”. In contrast to this fact, the solution of the dual problem is unique (see the discussion of Section 2.2). Moreover, the dual solution σ admits a clear physical or geometrical interpretation, for instance as a stress tensor or the normal to a surface. Hence, in the linear growth situation we wish to complete the above theorems by analogous regularity results for σ. C.1 Geometric problems of linear growth One of the most important (scalar) examples is the minimal surface case f (Z) = 1 + |Z|2 . A variety of references is available for the study of this variational integrand, let us mention the monographs of Giusti ([Giu2]) and Giaquinta/Modica/Souˇcek ([GMS2]) at this point. At first sight, ellipticity now is very bad since the inequalities in the second line of (3) just hold for the choices μ = 3 and q = 1. On the other hand, this rough estimate is not needed because it is possible to benefit from the geometric structure of the problem (see Remark 4.3). A class of integrands with this structure is studied, for instance, in [GMS1] following the a priori gradient bounds given in [LU2]. It turns out that in the minimal surface case ˆ generalized J-minimizers are of class C 1,α (Ω) and that we have uniqueness up to a constant. C.2 Linear growth problems without geometric structure The theory of perfect plasticity provides another famous variational integrand of linear growth. In this case the assumptions of smoothness and strict convexity imposed on f are no longer satisfied. Nevertheless, the example should be included in our discussion since we will benefit in Chapter 2 from the studies of Seregin ([Se1]–[Se6]) on this topic (compare the recent monograph [FuS2]). The quantity of physical interest is the stress tensor σ, which is only known to be partially regular (compare [Se4]). Even in the two-dimensional setting n = 2 we just have some additional information on the singular set (see [Se6]) and the model of plastic materials with logarithmic hardening (as described in B.1) serves as a regular approximation. It is already mentioned above that the vector-valued linear growth situation is covered by [AG2], provided that we restrict ourselves to smooth and strictly convex integrands. Anzellotti and Giaquinta prove Theorem 2 for genˆ eralized J-minimizers, hence the same regularity result turns out to be true for any u ∈ M (see Section 2.3.1 for details). It remains to study the properties of the dual solution which (as noted above) for linear growth problems is a quantity of particular interest.

1 Introduction

7

Before we summarize this brief overview in the table given below, we like to mention that of course there is a variety of further contributions where the class of admissible energy densities is equipped with some additional structure (see [AF4], [Lie2], [UU] and many others).

Some known regularity results in the convex case N =1

N >1

(1) DeGiorgi, Moser, Nash, A.1

Ladyzhenskaya/Ural’tseva ≤ ‘65

A.2

(1) 1 < p ≤ q < . . . Marcellini ≈ ‘90

(2) Anzellotti/Giaquinta ‘88 (3) Uhlenbeck, ‘77 (2) 2 ≤ p ≤ q < . . . Acerbi/Fusco ‘94, (3) bounded . . . , Choe ‘92 (3) n = 2: Frehse/Seregin ‘98 (2) n ≤ 4: Fuchs/Seregin ‘98

B.1 see N > 1

(2) Esposito/Mingione ‘00 (3) Mingione/Siepe ‘99

B.2

(1) μ < 1 + 2/n, q < . . . Fuchs/Mingione ‘00

C.1

(2) μ ≤ 4/n, “balanced” Fuchs/Osmolovskii ‘98 (3) [FM] (see N = 1)

(1)Jˆ Giaquinta/Modica/

—

Souˇcek ‘79 C.2

—

(2)Jˆ

[AG] (see A.1, N > 1)

(P)σ,pl Seregin ≈ ‘90 (1), (2), (3): Theorems 1–3, respectively ˆ (1)Jˆ, (2)Jˆ: corresponding results for generalized J-minimizers (P)σ,pl : partial regularity for the stress tensor in the theory of perfect plasticity

8

1 Introduction

In the following we are going to • have a close look at linear growth problems; • unify the results of A and B by the way including new classes of integrands; • discuss the substantial extensions which follow in cases A, B and C from a natural boundedness condition. Our main line skips from linear to superlinear growth and vice versa: in spite of the essential differences, these two items are strongly related by a non-uniform ellipticity condition (see Definition 3.4 and Assumption 4.1), by the applied techniques and to a certain extent by the obtained results. In particular, this relationship becomes evident while studying scalar variational problems with • mixed anisotropic linear/superlinear growth conditions. As the first center of interest, the discussion starts in Chapter 2 by considering the general linear growth situation. Here no uniqueness results for generalized minimizers can be expected and we concentrate on the dual solution σ which, according to the above remarks, is a reasonable physical point of view. The main contributions are i) uniqueness of the dual solution under very weak assumptions; ii) partial C 1,α -regularity for weak cluster points of J-minimizing sequences and, as a consequence, partial C 0,α -regularity for σ; iii) a proof of the duality relation σ = ∇f (∇a u∗ ) for a class of degenerate variational problems with linear growth. Here ∇a u∗ denotes the absolutely continuous part of ∇u∗ with respect to the Lebesgue measure. ad i). Standard arguments from convex analysis (compare [ET]) yield the uniqueness of the dual solution by assuming the conjugate function f ∗ to be strictly convex. We do not want to impose this condition since it is formulated in terms of f ∗ , hence there might be no easy way to check this assumption. In fact, using more or less elementary arguments, it is proved in Section 2.2 that there is no need to involve the conjugate function in an uniqueness theorem for the dual solution (see [Bi1]). ad ii). Following the lines of [GMS1], any weak cluster point u ∈ M ˆ associated to the original problem (see minimizes the relaxed problem (P) Appendix A.1). Alternatively (and as outlined in [BF1]), a local approach is preferred in Section 2.3.1 (see Remark 2.16 for a brief comment). In any case, the results of Anzellotti and Giaquinta apply and u is seen to be of class C 1,α on the non-degenerate regular set Ωu (see (23), Section 2.3). As a next step, the duality relation σ = ∇f (∇u∗ ), x ∈ Ωu∗ , is shown for a particular solution u∗ , hence σ is of class C 0,α on this set. ad iii). The duality relation is proved using local C 1,α -results for some u as above. As a consequence, information on the behavior of σ is only obtained on the u∗ -regular set. In Section 2.4, the almost everywhere identity ∗

1 Introduction

9

σ = ∇f (∇a u∗ ) is established for a class of degenerate problems which gives intrinsic regularity results in terms of σ (this is due to [Bi2]). Note that the applied technique completely differs from the previous considerations since we cannot rely on regularity results: arguments from measure theory are combined with the construction of local comparison functions (see Appendix B.3). Chapter 3 deals with the nearly linear and/or anisotropic situation. Here i) we introduce the notion of integrands with (s, μ, q)-growth; and give a unified and extended approach to ii) the results of type (1) and (3) outlined in the above table; iii) the corresponding theorems (2). Finally, reducing the generality of the previous sections, a theorem on iv) full C 1,α -regularity of solutions of two-dimensional vector-valued problems with anisotropic power growth completes Chapter 3. ad i). The main observation is clarified in Example 3.7. Three free parameters occurring in the structure and growth conditions imposed on the integrand f determine the behavior of solutions, which now uniquely exist in an appropriate energy class: the growth rate s of the integrand f under consideration, and the exponents μ, q of a non-uniform ellipticity condition. This leads to the notion of integrands with (s, μ, q)-growth which includes and extends the list given in A and B in a natural way. Note that related structure conditions for variational integrands with superquadratic growth are introduced in [Ma5]–[Ma7] (see Section 3.5 for a brief discussion). ad ii). Since regular solutions cannot be expected for the whole range of s, μ and q (we already mentioned [Gia2]), we impose the so called (s, μ, q)condition. Observe that we do not lose information in comparison with the known results (see Section 3.5). As a next step, uniform a priori Lqloc -estimates for the gradients of a regularizing sequence are proved. This enables us to apply DeGiorgi-type arguments with uniform local a priori gradient bounds as the result. The conclusion then follows in a well known manner (we refer to [BFM] for a discussion of scalar variational problems with (s, μ, q)-growth). It should be emphasized that the proof covers the whole scale of (s, μ, q)integrands without distinguishing several cases. ad iii). Here a blow-up procedure (compare [Ev], [CFM]) is used to prove partial regularity in the above setting (compare [BF2]). This generalizes the known results to a large extent (see Section 3.5).

10

1 Introduction

ad iv). With the higher integrability results of the previous sections it is possible (following [BF6]) to refer to a lemma due to Frehse and Seregin. In Chapter 4 we return to problems with linear growth, where we first benefit from some of the techniques outlined in Chapter 3, i.e. i) a regular class of μ-elliptic integrands with linear growth is introduced. Then the results are substantially improved by ii) studying bounded solutions (in some natural sense); iii) considering two-dimensional problems. We finish the study of linear growth problems by proving the iv) sharpness of the results. ad i). Example 3.9 also provides a class of μ-elliptic integrands with linear growth in the sense that for all Z, Y ∈ RnN − μ − 1 λ 1 + |Z|2 2 |Y |2 ≤ D2 f (Z)(Y, Y ) ≤ Λ 1 + |Z|2 2 |Y |2

(4)

holds for some μ > 1 and with constants λ, Λ. If μ < 1 + 2/n, then this class is called a regular one since generalized minimizers are unique up to a constant and since Theorems 1 and 3 for functions u ∈ M will be established following the arguments of Chapter 3 (see [BF3]). Let us shortly discuss the limitation μ < 1 + 2/n. Given a suitable regularization uδ , it is shown that 2−μ ωδ := 1 + |∇uδ |2 4 1 is uniformly bounded in the class W2,loc (Ω). This provides no information at all if the exponent is negative, i.e. if μ > 2. An application of Sobolev’s inequality, which needs the bound μ < 1 + 2/n, proves uniform local higher integrability of the gradients. The final DeGiorgi-type arguments will lead to the same limitation on the ellipticity exponent μ.

ad ii). The minimal surface integrand can be interpreted as a μ-elliptic example with limit exponent μ = 3 (recall that in the minimal surface case the regularity of solutions is obtained by using the geometric structure). Section 4.2 and [Bi4] are devoted to the question, whether the limit μ = 3 is of some relevance if the geometric structure condition is dropped. To this purpose some examples are discussed. Then, imposing a natural boundedness condition, we prove even in the vector-valued setting (without assuming f (Z) = g(|Z|2 )) that a generalized minimizer u∗ of class W11 (Ω; RN ) exists. Moreover, u∗ uniquely (up to a constant) determines the solutions of the problem f (∇w) dx + f∞ (u0 − w) ⊗ ν dHn−1 → min in W11 Ω; RN . (P ) Ω

∂Ω

1 Introduction

11

If, as a substitute for the geometric structure, μ < 3 is assumed, then the uniqueness of generalized minimizers up to a constant as well as Theorem 1 and Theorem 3 are true. As indicated above, the proof of i) does not extend to these results: in Sections 4.2.2.1 and 4.2.3 we do not differentiate the Euler equation, thus we avoid to use Sobolev’s inequality. Moreover, in the case μ < 3, a preliminary iteration gives uniform Lploc -gradient bounds for any p. This is the reason why we may use Hölder’s inequality and finally adjust the DeGiorgi iteration exponent to get the conclusion. ad iii). It turns out (compare [Bi5]) that a boundedness condition is superfluous to establish the results of ii) in the two-dimensional case n = 2 (with the usual structure in the vector-valued setting). Note that, once more, μ = 3 is exactly the limit case within reach. ad iv). Extending the ideas of [GMS1], an example is given which shows that the problem (P ) in general does not admit a W11 -solution if the ellipticity condition merely holds for some μ > 3. Since the energy density under consideration is explicitely depending on x, we have to show first in Section 4.2.2.2 (as a model case) that a smooth x-dependence does not affect the above mentioned theorems, thus our example really is a counterexample (see also [BF8]). Chapter 5 once more deals with the study of superlinear growth problems, where a boundedness condition analogous to Chapter 4.2 is supposed to be valid. We prove (in addition referring to [BF7], [BF9]) i) higher integrability and, as a corollary, a theorem of type (2) for variational integrands with a wide range of anisotropy. Then, as a model case, ii) scalar obstacle problems are studied for this class of energy densities and we prove a theorem of type (1). ad i). Recalling the ideas of Chapter 4 we expect that these techniques may be applied to improve the results of Section 3.3 and Section 3.4 for bounded solutions in the case of variational integrals with superlinear growth. If we consider integrands with anisotropic (p, q)-growth, then the corresponding relation between p and q should read as q < p+2. However, as proved in [BF5], 1 -solutions if q < p + 2/3. The the “linear growth techniques” just yield Wq,loc reason for this “lack of anisotropy” is the following: in Section 4.2 we could benefit from the growth rate 1 = q of the main quantity ∇f (Z) : Z under consideration. In the anisotropic superlinear case however, we just have the lower bound p < q of this quantity. This is the reason why we change methods again and give a refined study of an Ansatz which traces back to [Ch]. As a result, the full correspondence to the linear growth situation is established, i.e. with the assumption

12

1 Introduction

− μ q−2 λ 1 + |Z|2 2 |Y |2 ≤ D2 f (Z)(Y, Y ) ≤ Λ 1 + |Z|2 2 |Y |2 for all Z, Y ∈ RnN with positive constants λ, Λ and for exponents μ < 3, q > 1, higher local integrability follows from q < 4−μ. This provides (together with some natural hypothesis) a corollary on partial regularity. ad ii). Here, as a model case, we include the study of scalar obstacle prob1,α -regularity under the same lems. The methods as described in i) yield full Cloc condition q < 4 − μ which is quite weak (recall the counterexample of Section 4.4). Chapter 6 (see [Bi6]) closes the line with the consideration of • scalar variational problems with mixed anisotropic linear/superlinear growth conditions. Here, on one hand, we essentially have to rely on the wide range of anisotropy which is admissible on account of Chapter 5. On the other hand, a refined study of the dual problem is needed since a dual solution may even fail to exist. This is caused by a possible anisotropic behavior of the superlinear part itself. Nevertheless, we obtain locally regular and uniquely determined (up to a constant) generalized minimizers which in return provide a “local stress tensor”. We finish our studies with three appendices: the first one identifies the different ways to define generalized minimizers (recall Remark 1.1). The main Theorem A.6 (see [BF4]) proves, as a corollary, the uniqueness results applied in Chapter 4 which are based on the different approaches, respectively. In Appendix B some density results are collected, where either a rigorous proof is hardly found in the literature or the claims have to be adjusted to the situation at hand. Maybe, the construction of local comparison functions given in Section B.3 is the only result which is unknown to the reader (compare [BF1]). This helpful lemma is used several times studying linear growth problems. It is outlined in Appendix C (see [BF10], [ABF]) that the methods discussed throughout this monograph at least partially extend to the study of generalized Newtonian fluids. We did not include this material in the previous sections in order to keep the main line of the standard setting of the calculus of variations.

2 Variational problems with linear growth: the general setting

Following the main line sketched in the introduction, we start by considering the general linear growth situation. Recall that the variational problem (P) then may fail to have solutions which leads to suitable notions of generalized minimizers. On the other hand, it is quite natural to introduce the dual variational problem which is of particular interest in the setting of this chapter since no uniqueness results on generalized minimizers are available. Thus, we first have to give some introductory remarks on convex analysis in Section 2.1.1 in order to obtain a precise definition of the dual variational problem (P ∗ ). A first analysis of the dual solution(s) is given in Section 2.1.2: here a regularizing sequence is constructed which, in Lemma 2.6, is shown to converge to a maximizer of (P ∗ ). As a first regularity result, we prove that this maximizer is of class 1 (Ω; RnN ) (compare Section 2.1.3). W2,loc One essential motivation for the study of the dual variational problem is the uniqueness of solutions. In Section 2.2 such a uniqueness result for the dual solution σ is derived under very weak assumptions, in particular Theorem 2.15 does not depend on the strict convexity of the conjugate function. Two theorems of type (2) are outlined in the next section: each L1 -cluster ˆ (see point u∗ of a J-minimizing sequence solves some relaxed problem (P) Remark 2.16 and Appendix A.1) which, on account of [AG2], implies C 1,α regularity on the non-degenerate regular set Ωu∗ . We then establish the existence of u∗ as above such that the duality relation σ = ∇f (∇u∗ ) holds almost everywhere on Ωu∗ . This yields C 0,α -regularity of the dual solution σ on this set. In Section 2.4 we have a more detailed look at the degenerate situation: the above results for σ are formulated in terms of Ωu∗ , i.e. they involve the regular set of a special generalized minimizer. In order to obtain an intrinsic theory, we now prove that the duality relation in fact holds almost everywhere for a certain class of degenerate problems.


14


Throughout this chapter the variational integrand is supposed to satisfy the following general hypothesis: Assumption 2.1. The function f is smooth, strictly convex and of linear growth in the following sense: i) f ∈ C 2 RnN . ii) f (1 − λ)Z + λY < (1 − λ) f (Z) + λ f (Y ) for all Z = Y ∈ RnN and for all 0 < λ < 1. Suppose further that there is a positive number ν1 such that for all Z, Y ∈ RnN 1 0 ≤ D2 f (Z)(Y, Y ) ≤ ν1 |Y |2 . 2 1 + |Z| iii) There is a real number ν2 > 0 such that |∇f (Z)| ≤ ν2 for all Z ∈ RnN . iv) For numbers ν3 > 0 and ν4 ∈ R we have f (Z) ≥ ν3 |Z| + ν4 for all Z ∈ RnN . For the sake of simplicity the boundary values u0 under consideration are supposed to be of class W21 (Ω; RN ). As outlined in Remark 2.5, this restriction on the boundary data can easily be removed.

2.1 Construction of a solution for the dual problem 1 (Ω; RnN ) which is of class W2,loc We are going to give some introductory remarks on the dual problem associated to (P). Moreover, a suitable regularization is introduced in Section 2.1.2. As an immediate consequence we obtain in Section 2.1.3 a maximizer σ of the 1 problem (P ∗ ) which is of class W2,loc (Ω; RnN ). 2.1.1 The dual problem Here we recall some well known facts from convex analysis leading to the notion of the dual problem. As a reference one may choose, for instance, [Ro] or [Ze], we mostly follow the book of Ekeland and Temam ([ET]). Definition 2.2. Consider a Banach space V , its dual V ∗ and a function G : V → R. Then the polar or conjugate function of G is defined for all v ∗ ∈ V ∗ by G∗ (v ∗ ) := sup v, v ∗ − G(v) . v∈V

The bipolar function is given for all v ∈ V by G∗∗ (v) := sup v, v ∗ − G∗ (v ∗ ) . v ∗ ∈V ∗

1 2.1 Construction of a dual solution of class W2,loc

15

Since we always consider lower semicontinuous and convex functions G, the bipolar function satisfies (see [ET], Prop. 4.1, p. 18) G∗∗ (v) = G(v)

for all v ∈ V .

(1)

If the subdifferential of G is denoted by ∂G (see [ET] pp. 20) and if ∂G(v) = ∅, then we have the duality relation: v ∗ ∈ ∂G(v) ⇔ G(v) + G∗ (v ∗ ) = v, v ∗ . This gives for our smooth integrand f : RnN → R: f (w) + f ∗ ∇f (w) = w : ∇f (w) .

(2)

Here and in what follows the symbol Z : Y is used to denote the standard scalar product in RnN . We next derive an alternative expression for J[w], w ∈ W11 (Ω; RN ): given f as above, we consider the functional G : L1 (Ω; RnN ) → R, f (p) dx for all p ∈ L1 (Ω; RnN ) . G(p) := Ω

Then Proposition 2.1, [ET], p. 271, can be applied, hence, together with (1) and the definition of the conjugate function, we see that

f (p) dx = G(p) = G∗∗ (p) Ω

∗ κ : p dx − G (κ) = sup κ∈L∞ (Ω;RnN ) Ω

∗ κ : p dx − f (κ) dx . = sup κ∈L∞ (Ω;RnN )

Ω

Ω

This formula holds for all p ∈ L1 (Ω; RnN ), in particular for p = ∇w, w ∈ W11 (Ω; RN ). We obtain the representation formula

∗ J[w] = sup κ : ∇w dx − f (κ) dx . (3) κ∈L∞ (Ω;RnN )

Ω

Ω

Remark 2.3. Using the notation introduced by Ekeland and Temam, Chapter III.4, pp. 58, we arrive at (3) if we set J[w] = G(w, Λw) and Φ(w, p) = J(w, Λw − p), where the linear operator Λ is the ∇-operator. The representation formula (3) motivates to define the Lagrangian l(w, κ) for ◦

all (w, κ) = (u0 + ϕ, κ) in the class (u0 + W11 (Ω; RN )) × L∞ (Ω; RnN ) by the formula ∗ κ : ∇w dx − f (κ) dx = l(u0 , κ) + κ : ∇ϕ dx . l(w, κ) := Ω

Ω

Ω

16


Now, the dual functional R: L∞ (Ω; RnN ) → R is given by R[κ] :=

l(w, κ) ,

inf ◦

w∈u0 +W11 (Ω;RN )

and the dual problem reads as: to maximize R among all functions κ ∈ L∞ (Ω; RnN ) .

(P ∗ )

Remark 2.4. i) The definition of R shows that we have for any κ ∈ L∞ (Ω; RnN ) ⎧ ⎨ −∞ if div κ = 0 , R[κ] = ⎩ l(u , κ) if div κ = 0 . 0 ii) On the complement of the set Im(∇f ) we have the identity f ∗ ≡ +∞, thus for any κ under consideration we may assume that κ(x) ∈ Im(∇f ) almost everywhere. Let us finally mention one essential property of the dual functional: inf

J[w] =

◦

w∈u0 +W11 (Ω;RN )

sup

R[κ] .

(4)

κ∈L∞ (Ω;RnN )

Here the estimate J[w] =

sup κ∈L∞ (Ω;RnN )

=

sup

l(w, κ) ≥

sup

inf ◦

κ∈L∞ (Ω;RnN ) v∈u0 +W1 (Ω;RN )

l(v, κ)

1

R[κ]

κ∈L∞ (Ω;RnN )

is obvious, for the opposite inequality we either refer to [ET] or to the following Lemma 2.6, where the inf-sup relation is proved as a byproduct (compare Remark 2.9, i)). 2.1.2 Regularization Approximating the original problem in a well known way (compare, e.g. [Se4]), a special maximizing sequence for the dual problem is constructed in this subsection. Here we have to recall that the boundary values u0 are assumed to be of class W21 (Ω; RN ). The problem (P) is approximated in the following way: for any 0 < δ < 1 we consider the functional ◦ δ |∇w|2 dx + J[w] , w ∈ u0 + W21 Ω; RN , Jδ [w] := 2 Ω


17

and denote by uδ the unique solution of Jδ [w] → min ,

◦ w ∈ u0 + W21 Ω; RN .

(Pδ )

Setting fδ :=

δ 2 | · | + f , τδ := ∇f (∇uδ ) , σδ := δ∇uδ + τδ = ∇fδ (∇uδ ) , 2

we have the Euler equation σδ : ∇ϕ dx = 0

◦

for all ϕ ∈W21 (Ω; RN ) .

(5)

Ω

The minimality of uδ implies Jδ [uδ ] ≤ Jδ [u0 ] ≤ J1 [u0 ], hence there are positive constants c1 , c2 such that 2 δ |∇uδ | dx ≤ c1 , f (∇uδ ) dx ≤ c2 . (6) Ω

Ω

Remark 2.5. If we consider boundary values of class W11 (Ω; RN ), then the above regularization has to be applied to an approximating sequence {um 0 } ⊂ ∞ N 1 N C (Ω; R ) of boundary values converging in W1 (Ω; R ) to u0 . As a result, the regularized sequence depends on m, i.e. uδ = um δ . It will turn out in this chapter that this is no difference at all provided we have the uniform a priori bound (6). This, however, can be achieved by choosing δ = δ(m) sufficiently small. For details we refer to [Bi5]. The first inequality of (6) immediately gives 2 2 δ ∇uδ L2 (Ω;RnN ) = δ δ |∇uδ | dx → 0

as δ → 0 .

(7)

Ω

Since ∇f is bounded, we may also assume that τδ L∞ (Ω;RnN ) ≤ c . Passing to a subsequence (which is not relabeled) we obtain limits τ ∈ ∗ L∞ (Ω; RnN ) and σ ∈ L2 (Ω; RnN ) such that τδ τ in L∞ (Ω; RnN ) as well as σδ σ = τ

in L2 (Ω; RnN )

as δ → 0 .

Note that the convergence of a subsequence {σδ } yields by (7) the convergence of the corresponding subsequence {τδ } and vice versa. The following lemma shows that we have produced a maximizer of the dual variational problem. Lemma 2.6. i) Any weak L2 -cluster point σ of the sequence {σδ } is admissible in the sense that we have div σ = 0.

18


ii) Any weak L2 -cluster point σ of the sequence {σδ } maximizes the dual variational problem (P ∗ ). Remark 2.7. Note that a strict convexity condition for the dual function f ∗ is not imposed, i.e. the uniqueness of maximizers remains to be proved (compare Section 2.2). Remark 2.8. Lemma 2.6 corresponds to Lemma 2 of [Se4] where the case of integrands depending on the modulus of the gradient is considered. Similar results were obtained in [Se5], Lemma 3.2, and [Se6], Lemma 3.1. Proof of Lemma 2.6. Equation (5) yields for the sequence {σδ } under consideration τδ : ∇ϕ dx + δ ∇uδ : ∇ϕ dx = 0 for all ϕ ∈ C0∞ (Ω, RN ) . Ω

Ω

This implies i) by the above stated convergences. To complete the proof, observe that the duality relation (given in (2)) τδ : ∇uδ − f ∗ (τδ ) = f (∇uδ ) implies δ 2 τδ : ∇uδ − f ∗ (τδ ) dx |∇uδ | dx + Jδ [uδ ] = 2 Ω Ω δ =− σδ : ∇uδ − f ∗ (τδ ) dx |∇uδ |2 dx + 2 Ω Ω δ =− σδ : ∇u0 − f ∗ (τδ ) dx , |∇uδ |2 dx + 2 Ω Ω where we used (5) to get the last equation. We have ∞ nN sup R[κ] : κ ∈ L Ω; R ◦ ≤ inf J[w] : w ∈ u0 + W11 (Ω; RN ) ≤ J[uδ ] ≤ Jδ [uδ ] δ 2 σδ : ∇u0 − f ∗ (τδ ) dx |∇uδ | dx + = − 2 Ω Ω δ τδ : ∇u0 − f ∗ (τδ ) dx |∇uδ |2 dx + = − 2 Ω Ω +δ ∇uδ : ∇u0 dx . Ω

(8)

Passing to the limit, using the upper semicontinuity of − Ω f ∗ (·) dx with respect to weak-∗ convergence and observing that the last integral on the right-hand side of (8) tends to 0 as δ → 0, we obtain |∇uδ |2 dx → 0 as δ → 0 (9) δ Ω

as well as the maximality of σ.


19

Remark 2.9. i) Replacing the first inequality in (8) by a strict one, the inf-sup relation (4) follows by contradiction. ii) Inequality (8) proves in addition that {uδ } is a J-minimizing sequence. 1 -regularity for the dual problem 2.1.3 W2,loc

The first regularity result for the dual problem reads as Theorem 2.10. Let σ denote a weak L2 -cluster point of the sequence {σδ }. Then we have 1 (Ω; RnN ) . σ ∈ W2,loc Remark 2.11. Again we benefit from arguments outlined in [Se2] and [Se4]– [Se6]. Proof of Theorem 2.10. We fix a converging sequence {σδ }. Using the standard difference quotient technique, it is easily seen that uδ (recall the notation of 2 (Ω; RN ). Moreover, since |D2 fδ | is bounded, Section 2.1.2) is of class W2,loc 1 ∇fδ (∇uδ ) is of class W2,loc (Ω; RnN ) with partial derivatives (almost everywhere) ∂γ ∇fδ (∇uδ ) = D2 fδ (∇uδ ) ∂γ ∇uδ , · , γ = 1, . . . , n . Now, given ϕ ∈ C0∞ (Ω; RN ), we take ∂γ ϕ, γ = 1, . . . , n, as an admissible choice in the Euler equation (5). An integration by parts implies together with the above remarks D2 fδ (∇uδ ) ∂γ ∇uδ , ∇ϕ dx = 0 for all ϕ ∈ C0∞ (Ω; RN ) . (10) Ω

Using standard approximation arguments, (10) is seen to be true for all ϕ ∈ W21 (Ω; RN ) which are compactly supported in Ω. In particular, if some ball Br (x0 ) Ω is fixed, then ϕ = ∂γ uδ η 2 , η ∈ C0∞ (Ω), 0 ≤ η ≤ 1, η ≡ 1 on Br (x0 ), is admissible in (10) and we obtain

D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ η 2 dx Ω = −2 D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ uδ ⊗ ∇η η dx =: I2 .

I1 :=

(11)

Ω

Here we always take the sum with respect to γ = 1, . . . , n. An upper bound for |I2 | is given by (compare Assumption 2.1 and (6))

20

2 Variational problems with linear growth: the general setting 1 2

12 D fδ (∇uδ ) ∂γ uδ ⊗ ∇η, ∂γ uδ ⊗ ∇η dx

|I2 | ≤ c I1

2

Ω 1 2

≤ c(∇η) I1

12 1 − 1 + |∇uδ |2 2 + δ |∇uδ |2 dx

(12)

Ω 1

≤ c I12 . Now, the Cauchy-Schwarz inequality shows that we have almost everywhere |∇σδ |2 = D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ σδ 12 2 12 2 ≤ D fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ . D fδ (∇uδ ) ∂γ σδ , ∂γ σδ Since |D2 fδ | is uniformly bounded, there exists a constant, independent of δ, such that (13) |∇σδ |2 ≤ c D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ . Combining (11)–(13) we have proved that the sequence {σδ } is uniformly 1 (Ω; RnN ). This, together with the weak (with respect to δ) bounded in W2,loc convergence of σδ , yields the theorem.

2.2 A uniqueness theorem for the dual problem We now concentrate on the uniqueness of the dual solution which usually is established by assuming the conjugate function f ∗ to be strictly convex (see [ET], (3.34), p. 146). This hypothesis is formulated in terms of the conjugate function, hence, it might be difficult to verify the assumption for a given class of integrands f . Here we show by means of more or less elementary arguments that the strict convexity, the smoothness and the linear growth of f in the sense of Assumption 2.1 are sufficient to imply the uniqueness of the dual solution (without additional restrictions). The main idea is to construct (using Theorem 2.10) one special maximizer σ of the dual problem which is almost everywhere seen to be a mapping into the open set Im(∇f ). On this set f ∗ is known to be strictly convex by strict convexity of f . Thus, there is one solution σ for which we do not have to care about the fact that f ∗ on the closure of Im(∇f ) might not be strictly convex. This will give our uniqueness result. Remark 2.12. Alternatively, Corollary 2.18 together with the duality relation (31) of Section 2.3.2 could be used to provide one maximizer with values in the open set Im(∇f ) . However, it should be emphasized that Theorem 2.15 also covers the degenerate case in the sense that D2 f (Z)(Y, Y ) = 0 is not excluded.

2.2 A uniqueness theorem for the dual problem

21

Theorem 2.13 below is the main tool to prove the uniqueness of the dual solution. Here and in the following we let U := Im(∇f ). Theorem 2.13. Any weak L2 -cluster point σ of the δ-regularization given in Section 2.1.2 satisfies x ∈ Ω : σ(x) ∈ ∂U = 0 . Here | · | denotes the Lebesgue measure Ln . For the proof we need the following observation. Lemma 2.14. For all real numbers K > 0 there is an ε > 0 such that for all Z ∈ RnN dist ∇f (Z), ∂U < ε ⇒ |Z| > K . Proof of Lemma 2.14. Note that, by the strict convexity of f , Z = Y ∈ RnN implies ∇f (Z) − ∇f (Y ) : (Z − Y ) > 0 , i.e. ∇f is a one-to-one mapping. In fact, we have ∇f (Z) − ∇f (Y ) : (Z − Y ) 1 D2 f sZ + (1 − s)Y (Z − Y ), (Z − Y ) ds ≥ 0 . = 0

Setting g(s) := f sZ+(1−s)Y , equality would give g (s) ≡ 0 for all s ∈ (0, 1) which contradicts the strict convexity. Now fix a real number K > 0. Since ∇f is continuous and one-to-one we may apply the Theorem on Domain Invariance (compare [Sch], Corollary 3.22, p. 77) to see that U is an open set. Thus ∇f BK (0) U and there is an ε = ε(K) such that dist ∇f BK (0) , ∂U > ε . This proves the lemma. Proof of Theorem 2.13. With the notation of Section 2.1.2 we consider a sequence δm → 0 as m → ∞ such that the weak L2 -limit σ of {σδm } exists. Suppose by contradiction that there is a real number γ > 0 and a set Σ ⊂ Ω satisfying |Σ| > γ and σ(x) ∈ ∂U for all x ∈ Σ . Here and in the following, sets of Lebesgue measure zero are neglected. If necessary, we always choose suitable subsequences from our given sequence {δm } – omitting further indices – such that all the limits below are well defined.

22


Since the approximating sequence σδ is uniformly bounded in the Sobolev 1 (Ω, RnN ) (compare Theorem 2.10) we may assume that class W2,loc σδ (x) → σ(x)

for all x ∈ Σ as δ → 0 .

Because of (7), Section 2.1.2, in addition δ ∇uδ (x) → 0

for all x ∈ Σ as δ → 0

can be assumed. Moreover, Egoroff’s theorem yields a measurable set E ⊂ Ω such that γ |Ω − E| < 2 and such that σδ ⇒ σ

δ ∇uδ ⇒ 0

as well as

on E as δ → 0 .

(14)

The set E is measurable and |Σ| = |Σ ∩ E| + |Σ − E| immediately implies

γ . (15) 2 By construction, {uδ } is a J-minimizing sequence (see Remark 2.9, ii)). In particular, on account of the linear growth of f , there is a real number c0 > 0 such that for all δ sufficiently small |∇uδ | dx < c0 . (16) |Σ ∩ E| >

Ω

Given c0 we choose K > 2c0 /γ and 0 < ε = ε(K) as determined in Lemma 2.14. The uniform convergence (14) shows that for all x ∈ E and for all δ sufficiently small |σδ (x) − σ(x)|
Σ∩E

>

K dx Σ∩E

γ K > c0 . 2

Thus, the theorem is proved. Theorem 2.15. With the above Assumption 2.1 on f the dual problem (P ∗ ) admits a unique solution σ. Proof. We fix a weak L2 -limit σ of the δ-approximation. By Lemma 2.6 this limit is known to be a solution of (P ∗ ). Suppose by contradiction that the dual problem admits a second maximizer σ ˜ = σ. Now, Ω is divided into four parts, Ω = Σ 0 ∪ Σ 1 ∪ Σ2 ∪ Σ3 , where we have by definition σ(x) = σ ˜ (x)

for x ∈ Σ0 ,

σ(x) = σ ˜ (x) , σ(x) ∈ U , σ ˜ (x) ∈ U

for x ∈ Σ1 ,

σ(x) = σ ˜ (x) , σ(x) ∈ U , σ ˜ (x) ∈ ∂U for x ∈ Σ2 , |Σ3 | = 0 . The last equality is justified by Theorem 2.13 and by Remark 2.4, ii). The conjugate function f ∗ of f is a lower semicontinuous convex function on U and by (2) of Section 2.1.1 it is also seen to be strictly convex on U . This implies |Σ1 | = 0 . (17) In fact, setting σ(x) + σ ˜ (x) , 2 the convexity of f ∗ gives for almost every x ∈ Ω κ(x) :=

1 1 ˜ (x) , f ∗ κ(x) ≤ f ∗ σ(x) + f ∗ σ 2 2 and the strict inequality holds for all x ∈ Σ1 . Recalling Remark 2.4, i), we obtain

24


R[κ] = l(u0 , κ) Ω

σ+σ ˜ 2

Ω

σ+σ ˜ 2

= ≥ =

: ∇u0 dx −

∗

f ∗ (κ) dx

f (κ) dx − Ω−Σ1

1 : ∇u0 dx − 2

Σ1

1 f ∗ (σ) dx − 2 Ω

(18) f ∗ (˜ σ ) dx

Ω

1 1 R[σ] + R[˜ σ ] = sup R . 2 2

Since the strict inequality would hold for |Σ1 | > 0, assertion (17) is proved by contradiction. We next claim that |Σ2 | = 0 .

(19)

To prove (19), we fix x ∈ Σ2 , i.e. P := σ(x) ∈ U

and

Q := σ ˜ (x) ∈ ∂U .

Now, U is an open set and Theorem 6.1, [Ro], p. 45, proves [P, Q) := {(1 − t) P + t Q : 0 ≤ t < 1} ⊂ U .

(20)

If we set for m ∈ N sufficiently large m−1 (Q − P ) , m 1 m−1 := P + (Qm − P ) = P + (Q − P ) , 2 2m

Qm := P + Rm

and if we introduce for t ∈ (0, 1) the function g(t) = f ∗ P + t (Q − P ) which, by (20), is known to be strictly convex, then we see (compare, for instance, the proof of Theorem 4.4, [Ro], p. 26) that there is a real number 0 < ε = ε(P, Q) such that ∗

∗

f (Rm ) − f (P )

m−1 2m

=

g (s) ds ≤ g

0

∗

∗

f (Qm ) − f (Rm ) =

m−1 m m−1 2m

g (s) ds ≥ g

m−1 2m m−1 2m

m−1 −ε, 2m m−1 +ε. 2m

Combining these inequalities it is proved that f ∗ (Rm ) ≤

1 ∗ 1 f (P ) + f ∗ (Qm ) − ε . 2 2

(21)

2.3 Partial C 1,α - and C 0,α -regularity . . .

25

Since a lower semicontinuous convex function is continuous on straight lines to the boundary and since ε is not depending on m, we may pass in (21) to the limit m → ∞ and obtain for all x ∈ Σ2 1 σ(x) + σ ˜ (x) 1 < f ∗ (σ(x)) + f ∗ (˜ f ∗ κ(x) = f ∗ σ (x)) . 2 2 2 The same computations as outlined in (18) give (19), we have shown Ω = Σ0 ∪ (Ω − Σ0 )

with

|Ω − Σ0 | = 0 ,

and the theorem is proved by the definition of Σ0 .

2.3 Partial C 1,α - and C 0,α -regularity, respectively, for generalized minimizers and for the dual solution In the general setting of vector-valued variational problems with linear growth full regularity cannot be expected – even if we additionally assume that D2 f (Z) > 0 holds for any matrix Z ∈ RnN . We prove in Section 2.3.1 that C 1,α -regularity of generalized minimizers u∗ ∈ M =

u ∈ BV Ω; RN : u is the L1 -limit of a J-minimizing ◦ sequence from u0 + W11 (Ω; RN )

holds on the non-degenerate regular set Ωu∗ . To give a precise definition of this set, we observe that for almost all x ∈ Ω there exists a matrix P ∈ RnN such that 1 1 ∗ lim |∇u − P | := lim |∇a u∗ − P | dx r→0 |Br (x)| B (x) r→0 |Br (x)| B (x) r r 1 + lim |∇s u∗ | = 0 . (22) r→0 |Br (x)| B (x) r Here ∇a u∗ denotes the absolutely continuous part of ∇u∗ with respect to the Lebesgue measure, whereas ∇s u∗ is used as the symbol for the singular part. Then, the non-degenerate regular set is defined for u∗ ∈ M via Ωu∗ := x ∈ Ω : (22) holds with D2 f (P ) > 0 . (23) The main Theorem to study the smoothness of generalized minimizers is given in [AG2] where local minimizers of some relaxed functional are considered. In order to apply this theorem to u∗ ∈ M as given above we like to remark

26


Remark 2.16. One possibility is to follow the lines of [GMS1] and to work in the space BVu0 (Ω; RN ). This is outlined in Appendix A.1. We prefer a local approach which is based on the construction of suitable comparison functions as given in Appendix B.3. It will turn out in addition that this local point of view provides a helpful tool for the consideration of degenerate problems (compare Section 2.4). Another application of Lemma B.5 is found in Section 4.3. In Section 2.3.2 it remains to prove the existence of u∗ ∈ M such that σ = ∇f (∇u∗ ) holds almost everywhere on Ωu∗ . This gives the C 0,α -regularity of the dual solution on this set. In particular, if we consider non-degenerate problems then the stress tensor is Hölder continuous on an open set of full measure. 2.3.1 Partial C 1,α -regularity of generalized minimizers Theorem 2.17. Suppose that the integrand f satisfies the general Assumption 2.1. Moreover, consider a J-minimizing sequence {um } from the affine class ◦

u0 + W11 (Ω; RN ) and u∗ ∈ L1 (Ω; RN ) satisfying um → u∗

in L1 (Ω; RN )

as m → ∞ .

If Ωu∗ is given according to (23) then Ωu∗ is an open set and we have u∗ ∈ C 1,α (Ωu∗ ; RN )

for any α ∈ (0, 1) .

Of course Theorem 2.17 implies Corollary 2.18. With the notation and with the assumptions of Theorem 2.17 let us suppose that we have in addition to Assumption 2.1 0 < D2 f (Z)(Y, Y )

for all Z, Y ∈ RnN , Y = 0 .

Then there exists an open set Ω0 of full measure, i.e. |Ω − Ω0 | = 0, such that u∗ ∈ C 1,α (Ω0 ; RN )

for any α ∈ (0, 1) .

Before proving Theorem 2.17 we introduce (following for example [GS], [AD] or [Bu]) a relaxed functional Jˆ on the space of functions of bounded variation and recall some well known properties (see Appendix A.1 for a more intensive discussion). The theorem then will be established as an immediate consequence of Lemma B.5 and Theorem 1.1 of [AG2] (compare Remark 2.16). ˆ always denotes a bounded Lipschitz domain. Here and below Ω ˆ RN ) the functional J[w; ˆ Ω] ˆ is given by Definition 2.19. For all w ∈ BV (Ω;

1 N 1 N ˆ R ), wk → w in Lloc (Ω; ˆ R ) . ˆ Ω] ˆ := inf lim inf J[wk ] : wk ∈ C (Ω; J[w; k→∞


27

The following properties of Jˆ are needed in our context: Proposition 2.20. ˆ RN )i) The functional Jˆ is lower semicontinuous with respect to L1loc (Ω; convergence. ˆ RN ). ˆ Ω] ˆ and J ˆ coincide on W 1 (Ω; ii) The functionals J[·, 1 |Ω iii) With the notation of Theorem 2.17, we have ◦ ˆ ∗ ; Ω] ≤ inf J[u] : u ∈ u0 + W11 Ω; RN . J[u Proof. Using L1 -approximations for BV-functions, the first part immediately ˆ RN ) ˆ The lower semicontinuity of J on W 1 (Ω; follows from the definition of J. 1 with respect to the L1loc -topology (see [AD]) implies part (ii.). To prove the third statement, consider a sequence {um } and u∗ as in Theorem 2.17. Here we assume without loss of generality that um ∈ C 1 (Ω; RN ) for all m ∈ N. The strong L1 -convergence um → u yields ˆ ∗ ; Ω] = inf J[u

lim inf J[wk ] : wk ∈ C 1 Ω; RN , k→∞

wk → u

∗

in

L1loc

Ω; R

N

◦ ≤ lim inf J[um ] = inf J[u] : u ∈ u0 + W11 Ω; RN , m→∞

and the proposition is proved. A deeper result is the following representation formula of Goffman and Serrin (see [GS]): Proposition 2.21. The representation formula s ∇ u a ˆ Ω] ˆ = f (∇ u) dx + f∞ d|∇s u| J[u, s |∇ u| ˆ ˆ Ω Ω ˆ RN ) where f∞ is the recession function of f defined holds for all u ∈ BV (Ω; by f (tX) . f∞ (X) = lim sup t t→+∞ As above, the absolutely continuous part of ∇u with respect to the Lebesgue measure is denoted by ∇a u, the singular part by ∇s u and ∇s u/|∇s u| is the Radon-Nikodym derivative. For a proof we also refer to [AD], where f is only required to be quasiconvex. The next proposition follows from [AG2], Theorem 2.1 and Proposition 2.2, respectively (see also [GMS1] and [Re]). ˆ RN ) and that there is a Proposition 2.22. Suppose that there is u ∈ BV (Ω; ˆ RN ) such that as m → ∞: sequence {um } ⊂ W11 (Ω;

28


ˆ RN ); i) um → u in L1 (Ω; 2 1 + |∇um | dx → 1 + |∇u|2 . ii) ˆ Ω

ˆ Ω

Then Jˆ is continuous with respect to this kind of convergence, i.e. ˆ → J[u; ˆ Ω] ˆ m ; Ω] ˆ J[u

as m → ∞ .

We now come to the Proof of Theorem 2.17. Consider a sequence {um } ⊂ ◦

u0 + W11 (Ω; RN ) such that ◦ 1 N J[um ] → inf J[u] : u ∈ u0 + W1 Ω; R and

um → u ∗

in L1 (Ω; RN )

as m → ∞ .

We improve the properties of {um } by using Lemma B.5, i.e. given x0 ∈ Ω we choose a ball BR (x0 ), B2R (x0 ) Ω and we choose a sequence {wm } ⊂ ◦

u0 + W11 (Ω; RN ) such that: i) wm → u∗ in L1 (Ω; RN ) as m → ∞; ii) lim J[wm ] ≤ lim J[um ]; m→∞

m→∞

iii) wm |∂BR (x0 ) = u∗|∂BR (x0 ) , where the traces are well defined functions of class L1 (∂BR (x0 ); RN ). We then let 1 N → R , I[w] := f (∇w) dx, • I : W1 BR (x0 ); R B (x ) 0 R • K := w ∈ W11 BR (x0 ); RN : w|∂BR (x0 ) = u∗|∂BR (x0 ) , and claim that (using the notation vm = wm |BR (x0 ) ) inf I = lim inf I[vm ] . K

m→∞

(24)

In fact, we argue by contradiction and assume that there is w ∈ K satisfying I[w] ≤ I[vm ] − δ for some δ > 0 and for all m sufficiently large. By the definition of K, the function ⎧ ⎨ w on B (x ) R 0 w ˜m := ⎩ w on the rest of Ω m is of class W11 (Ω; RN ) and, as a direct consequence, we have


29

lim sup J[w ˜m ] ≤ lim J[wm ] − δ . m→∞

m→∞

This provides a contradiction to ii), hence we have (24). Now we complete the proof of Theorem 2.17: to this purpose consider u ˜ ∈ BV (BR (x0 ); RN ) such that spt(˜ u − u∗ ) BR (x0 ). It is well known (and for the reader’s convenience we give a proof in Lemma B.1) that we may choose ˜m → u ˜ in L1 (BR (x0 ); RN ), u ˜m ∈ W11 (BR (x0 ); RN ), m ∈ N, satisfying u 2 1 + |∇˜ um | dx → 1 + |∇˜ u|2 BR (x0 )

BR (x0 )

and u ˜m|∂BR (x0 ) = u ˜|∂BR (x0 ) . Then we have u ˜m ∈ K, hence, by Proposition 2.20, by Proposition 2.22 and by (24) ˆ um ; BR (x0 )] = lim I[˜ ˆ u; BR (x0 )] = lim J[˜ um ] J[˜ m→∞

m→∞

(25)

ˆ ∗ ; BR (x0 )] , ≥ inf I = lim inf I[vm ] ≥ J[u K

m→∞

where the last inequality follows from vm → u∗ in L1 (Ω; RN ) (see i)) and from ˆ BR (x0 )]. Quoting Theorem 1.1 of [AG2] we have proved the definition of J[·; Theorem 2.17. Remark 2.23. In particular, the conclusion of Theorem 2.17 holds for each L1 -limit of the δ-regularization given in Section 2.1.2. 2.3.2 Partial C 0,α -regularity of the dual solution Now the second partial regularity result is proved, i.e. we concentrate on the dual solution σ. We prefer to give a “direct” proof just relying on our δregularization, an alternative way is outlined in [SE1/4/5/6]: by using the relaxed minimax inequality one can show that ∇u∗ = ∇f ∗ (σ) holds on the regular set of any cluster point u∗ of a J-minimizing sequence. As a consequence, we have σ = ∇f (∇u∗ ) which implies Theorem 2.24. Theorem 2.24. Suppose that we have Assumption 2.1 and let u∗ denote a weak cluster point of the δ-regularization introduced in Section 2.1.2. Moreover, consider the solution σ of the dual variational problem (P ∗ ). Then σ ∈ C 0,α (Ωu∗ ; RnN )

for any 0 < α < 1 ,

where Ωu∗ is the open set given above. Again we immediately obtain Corollary 2.25. If in addition to the assumptions of Theorem 2.24 0 < D2 f (Z)(Y, Y )

for all Z, Y ∈ RnN , Y = 0

is assumed, then partial C 0,α -regularity of σ follows on an open set of full measure.

30


Proof of Theorem 2.24. Consider the δ-regularization introduced in Section 2.1.2 and fix a subsequence with L1 -cluster point u∗ . Recalling σδ := δ ∇uδ + ∇f (∇uδ ) we have the equation σδ : ∇ϕ dx = 0 for all ϕ ∈ C01 (Ω; RN ) . (26) Ω

By Theorem 2.17 we know that u∗ satisfies u∗ ∈ C 1,α (Ωu∗ ; RN ) and that Ωu∗ is an open set. Now, for every open set G such that G ⊂ Ωu∗ we have ∇u∗ ∈ L∞ (G; RnN ) and additionally ∇f (∇u∗ ) : ∇ϕ dx = 0 for all ϕ ∈ C01 (G; RN ) . (27) G

ˆ by (25) u∗ is a This is the Euler equation with respect to the relaxation J: ˆ BR (x)] for a suitable ball around each point x ∈ Ω. local minimizer of J[·, The representation formula from Proposition 2.21 then shows that variations on the regular set imply (27). Combining the Euler equations (26) and (27) we obtain (passing to a subsequence) ∇uδ (x) → ∇u∗ (x)

for almost every x ∈ G as δ → 0.

(28)

In fact, inserting η 2 [uδ − u∗ ], η ∈ C01 (G), 0 ≤ η ≤ 1, and subtracting the equations we get σδ − ∇f (∇u∗ ) : ∇ η 2 [uδ − u∗ ] dx = 0 . G

The definition of σδ yields

∇f (∇uδ ) − ∇f (∇u∗ ) : (∇uδ − ∇u∗ ) η 2 dx G δ ∇uδ : ∇uδ − ∇u∗ η 2 dx + G σδ : [uδ − u∗ ] ⊗ ∇η η dx = −2 G +2 ∇f (∇u∗ ) : [uδ − u∗ ] ⊗ ∇η η dx . G

Now, ∇f (∇u∗ ) ∈ L∞ (Ω; RnN ) and (uδ − u∗ ) → 0 in L1 (Ω; RnN ), hence ∇f (∇u∗ ) : [uδ − u∗ ] ⊗ ∇η η dx → 0 as δ → 0 . G

Moreover,

G

δ ∇uδ : (∇uδ − ∇u∗ ) η 2 dx → 0

as δ → 0

(29)


31

follows from δ∇uδ 0 in L2 (Ω; RnN ) and δ Ω |∇uδ |2 dx → 0 (compare (9), Section 2.1.2). The first integral on the right-hand side of (29) can be written in the following form −2

δ∇uδ : [uδ − u∗ ] ⊗ ∇η η dx G −2 ∇f (∇uδ ) : [uδ − u∗ ] ⊗ ∇η η dx =: I1 + I2 . G

As above we immediately get I2 → 0 as δ → 0 and the same is true for I1 . This implies ∇f (∇uδ ) − ∇f (∇u∗ ) : (∇uδ − ∇u∗ ) η 2 dx → 0 as δ → 0 . G

Passing to a subsequence we get ∇f (∇uδ ) − ∇f (∇u∗ ) : (∇uδ − ∇u∗ ) → 0 almost everywhere in G (30) as δ → 0. For the sake of completeness we next briefly prove a remark which is needed to deduce (28) from (30). Remark 2.26. For any Z ∈ RnN we have lim ∇f (Y ) − ∇f (Z) : (Y − Z) = ∞ . |Y |→∞

Proof. Fix Z ∈ RnN . For any Y ∈ RnN the convexity of f implies

∇f (Y ) − ∇f (Z) : (Y − Z) ≥ f (Y ) − f (Z) − ∇f (Z) : (Y − Z) = f (Y ) − ∇f (Z) : Y + c(Z) ,

where c(Z) is a real number depending on Z. The definition of the conjugate function gives the generalized Young’s inequality ∇f (Z) : Y ≤ f ((1 − ε)Y ) + f ∗ (1 − ε)−1 ∇f (Z) for any real number 0 < ε < 1. By the open mapping theorem Im(∇f ) is an open set and we may choose ε > 0 sufficiently small such that in addition f ∗ (1 − ε)−1 ∇f (Z) < c(Z) can be assumed. Setting Q := (1−ε)Y it remains to estimate f (Y ) − f (1 − ε)Y ≥ εY : ∇f (1 − ε)Y = ≥

ε (f (Q) − f (0)) 1−ε

ε Q : ∇f (Q) 1−ε

32


which follows from the convexity of f . From Remark 2.26 and from (30) we deduce that ∇uδ (x) remains bounded for almost every x ∈ G, but then (30) immediately yields (28) by the strict convexity of f and we have ∇f ∇uδ (x) → ∇f ∇u∗ (x) for almost every x ∈ G as δ → 0 . Keeping (28) and ∇f (∇uδ ) σ in L2 (Ω; RnN ) as δ → 0 in mind, it is shown that in G . (31) σ(x) = ∇f ∇u∗ (x) This implies (31) for x ∈ Ωu∗ and the theorem is proved.

2.4 Degenerate variational problems with linear growth Let us recall the main line of the last section where we proved a smoothness result for the dual solution: the first step is to show that generalized minimizers u∗ ∈ M are regular on Ωu∗ (see Section 2.3.1). The essential tool to analyze the maximizer σ with the help of this information turns out to be the duality relation proved in (31) of Section 2.3.2 for all x ∈ Ωu∗ . (32) σ(x) = ∇f ∇u∗ (x) If degenerate problems are studied, i.e. if |Ω−Ωu∗ | > 0 has to be expected, then the approach outlined above in general does not lead to satisfying results. Let us sketch the two main problems by considering a prominent example (compare [GMS1]): f (Z) =

1 + |Z|

k

k1

,

k>2.

(33)

On one hand, (32) now is a quite vague statement since, due to the degeneracy of D2 f , the regular set Ωu∗ may be very small. On the other hand, and this is even more restrictive if we are interested in regularity results for σ, partial H¨ older continuity of σ for the (33) integrand ∗ 2 ∗ at hand follows a priori only on the set D f (∇u (x)) > 0 = ∇u (x) = 0 . However, an intrinsic theorem should be formulated in terms of σ, i.e. the domain of partial regularity is expected to be a ∗ σ(x) = 0 ⊃ ∇ u (x) = 0 . (34) Here the inclusion follows on account of (32) and of course is meant modulo sets of measure zero. Again ∇a u∗ denotes the absolutely continuous part of ∇u∗ with respect to the Lebesgue-measure and ∇s u∗ will be used as the symbol for the singular part.

2.4 Degenerate variational problems with linear growth

33

In this section, a generalization of the classical duality relation is established for almost all x ∈ Ω: we leave the regularity of u∗ as the starting point and prove by arguments from measure theory (in addition we use Appendix B.3) that there is a generalized minimizer u∗ ∈ M of the problem (P) such that for almost all x ∈ Ω . σ(x) = ∇f ∇a u∗ (x) Here, the degenerate situation D2 f ≥ 0 also is covered (compare Assumption 2.27). Coming back to the above example, we see that in fact equality (again modulo sets of measure zero) holds in (34) and analogous results of course are true in the case of more general degenerate integrands. As an application, an intrinsic regularity theorem independent of u∗ can be formulated just in terms of σ and the data (compare Section 2.4.2 for details). 2.4.1 The duality relation for degenerate problems In order to establish the duality relation (32) in the degenerate situation, the general Assumption 2.1 is refined by the additional Assumption 2.27. Assume in addition to Assumption 2.1 that i) f (Z) > 0 for all Z ∈ RnN , Z = 0, and f (0) = 0. ii) We have for all Z, Y ∈ RnN , Z = 0, Y = 0, 0 < D2 f (Z)(Y, Y ) . Remark 2.28. i) Of course i) is supposed without loss of generality and the model integrand considered in (33) satisfies ii). ii) The set Ωu∗ is characterized by 1 1 a ∗ lim |∇ u − P | dx + lim |∇s u∗ | = 0 r→0 |Br (x)| B (x) r→0 |Br (x)| B (x) r r for some matrix P ∈ RnN such that D2 f (P ) > 0 (see [AG2] and [BF1]). Hence, we have ∇a u∗ = 0 almost everywhere on the complement Ωcu∗ of Ωu∗ . However, this provides no results at all because no topological information on Ωcu∗ is available. Moreover, the singular part ∇s u∗ is not necessarily vanishing on Ωcu∗ . Let us recall the notion of the δ-regularization {uδ } introduced in Section 2.1.2. In particular, the convergence (7), Section 2.1.2, and Theorem 2.10 are needed to assume without loss of generality (after passing to a subsequence) i) ii)

σδ (x) → σ(x) for almost all x ∈ Ω , δ ∇uδ (x) → 0

for almost all x ∈ Ω ,

(35)

34


where σδ = δ∇uδ + ∇f (∇uδ ) and where σ denotes the unique solution of the dual variational problem (P ∗ ). Passing to another subsequence, if necessary, a L1 -cluster point u∗ of uδ is fixed in the following: L1 uδ −→: u∗ ∈ BV Ω, RN

as δ → 0 .

Then the main theorem of this section reads as follows. Theorem 2.29. The unique solution σ of the dual problem (P ∗ ) satisfies for almost all x ∈ Ω . σ(x) = ∇f ∇a u∗ (x) Remark 2.30. It remains an open question whether σ = ∇f (∇a u) holds for any generalized minimizer u ∈ M. For the proof of Theorem 2.29 we have to construct “large” sets of uniform convergence according to the following Proposition. Proposition 2.31. There is a measurable function v: Ω → RnN , and for any ε > 0 there is a compact set K Ω such that: / ∂Im(∇f ) for all x ∈ K; i) σδ ⇒ σ on K, σ(x) ∈ ii) δ∇uδ ⇒ 0 on K; iii) ∇uδ ⇒ v on K; iv) The restriction of v on K is a continuous function; v) |Ω − K| < ε. Remark 2.32. In the following it is obvious that we can restrict ourselves to the consideration of Lebesgue points of σ and ∇u∗ , respectively. This is always assumed as a general “hypothesis”. Proof of Proposition 2.31. Let us first define vji (x) := lim sup δ→0

∂ i u (x) , i ∈ {1, . . . , N } , j ∈ {1, . . . , n} , ∂xj δ

which by construction is a measurable function with values in R. Now fix ε > 0. The uniform convergence (as stated in i) and ii)) on a ˜ Ω with |Ω − K| ˜ compact set K < ε/2 follows on account of (35) and Egoroff’s theorem. Setting N = x ∈ Ω : σ(x) ∈ ∂Im(∇f ) it was proved in Section 2.2 that |N | = 0. Hence, we may choose an open set U ⊃ N with ˜ − U is a compact set such that i) and ii) are valid |U | < ε/2. Then K := K and such that in addition K satisfies v). Next observe that we have on K as δ → 0


∇f ∇uδ (x) = σδ (x) − δ ∇uδ (x) → σ(x) .

35

(36)

If x0 ∈ K is fixed, then σ(x0 ) ∈ / ∂Im(∇f ) implies that there is a constant ρ = ρ(x0 ) such that for all δ sufficiently small dist ∇f ∇uδ (x0 ) , ∂Im(∇f ) ≥ ρ . This means that we have for all δ sufficiently small ∇f ∇uδ (x0 ) ∈ C := Q ∈ Im(∇f ) : dist Q, ∂Im(∇f ) ≥ ρ . Since C is compact and since ∇f is a homeomorphism (compare the proof −1 −1 (C) is compact, in particular ∇f (C) is bounded of Lemma 2.14), ∇f and, as a consequence, lim sup |∇uδ (x)| < ∞ δ→0

for all x ∈ K .

(37)

Given (36) and (37), the pointwise convergence of ∇uδ (x) on K is clear since ∇f is one-to-one. Egoroff’s Theorem then proves iii) on a “large” compact ˜ ⊂ K, without loss of generality on K. The proof of iv) is an application set K of Lusin’s theorem and the proposition follows. We now come to the Proof of Theorem 2.29. Fix ε > 0 and choose K according to the Proposition 2.31. Since K is a measurable set, the LebesgueBesicovitch Differentiation Theorem yields |Br (x) ∩ K| = 1 r→0 |Br (x)| lim

for almost all x ∈ K .

(38)

It is also known that for almost all x ∈ Ω there exists a matrix P ∈ RnN such that 1 lim |∇u∗ − P | r→0 |Br (x)| B (x) r 1 1 a ∗ := lim |∇ u − P | dx + lim |∇s u∗ | = 0 . r→0 |Br (x)| B (x) r→0 |Br (x)| B (x) r r Let us first consider the case P = 0, i.e. x is a non-degenerate point. Going through the lines of Section 2.3 we observe that the duality relation as claimed in the theorem holds at this particular point x. Thus, we have to study the case 1 |∇u∗ | = 0 . (39) lim r→0 |Br (x)| B (x) r Observe that (39) implies 1 lim r→0 |Br (x)|

Br (x)

f ∇a u∗ dx = 0 .

(40)

36


In fact, on account of the continuity of f and since f (0) = 0 we may fix a real number λ > 0 and find κ > 0 such that |Z| < κ implies f (Z) < λ. We obtain

f ∇a u∗ dx Br (x) 1 ≤ lim sup f ∇a u∗ dx r→0 |Br (x)| Br (x)∩[|∇a u∗ |κ]

1 lim r→0 |Br (x)|

Here, the second term on the right-hand side vanishes by the linear growth of f and by (39): for some real numbers c1 , c2 > 0 we have 1 lim sup r→0 |Br (x)|

Br (x)∩[|∇a u∗ |>κ]

f ∇a u∗ dx

1 ≤ c1 lim sup |∇a u∗ | dx r→0 |Br (x)| Br (x)∩[|∇a u∗ |>κ] 1 +c2 lim sup 1 dx = 0 . r→0 |Br (x)| Br (x)∩[|∇a u∗ |>κ]

The remaining term is bounded from above by λ which is an arbitrary fixed positive number, and the claim (40) is proved. With (38) and (39) we define a set GK satisfying |K − GK | = 0: GK := x ∈ K : (38) and (39) are valid or σ(x) = ∇f ∇a u∗ (x) . Remark 2.33. Let us give a short comment on this definition. Consider the set of all x ∈ K such that (38) holds. Then we have to distinguish between the cases “P = 0”, i.e. (39) is true, and “P = 0”. As already mentioned, in the second case the duality relation is proved in Section 2.3.2, in particular |K − GK | = 0. According to this remark, we fix x ˆ ∈ GK satisfying (38) and (39) and recall x) the fact that f achieves its absolute minimum at Z = 0. Hence, 0 = ∇a u∗ (ˆ and 0 = ∇f (0). We claim that x) (41) σ(ˆ x) = 0 = ∇f (0) = ∇f ∇a u∗ (ˆ which immediately yields the theorem by passing to the limit ε → 0. To prove (41) assume by contradiction that σ(ˆ x) = 0. We now claim that there is a real number γ = γ(ˆ x) > 0 such that for all δ sufficiently small x)| . γ < |∇uδ (ˆ

(42)

To verify (42), let τ = |σ(ˆ x)| and choose δ0 > 0 sufficiently small. We obtain for all δ < δ0


|σδ (ˆ x) − σ(ˆ x)|
0 such that B4ρ0 (ˆ x) Ω and such that for any ρ < ρ0 γ ≤ |v(x)| 2

x) ∩ K . for all x ∈ Bρ (ˆ

Finally, setting κ = γ/4 and recalling the uniform convergence stated in Proposition 2.31, iii), we decrease δ0 , if necessary, and arrive at κ ≤ |∇uδ (x)|

for all x ∈ Bρ (ˆ x) ∩ K ,

0 < ρ < ρ0 ,

(44)

for all δ < δ0 . Remark 2.34. If in the sense of measures |∇uδ | |∇u∗ |

(45)

would be known, then the compactness of B ρ (ˆ x) ∩ K would imply κ |B ρ (ˆ x) ∩ K| ≤ lim sup |∇uδ | B ρ (ˆ x) ∩ K δ→0

≤ |∇u∗ | B ρ (ˆ x) ∩ K . Passing to the limit ρ → 0 a contradiction would follow from (39) and from the density relation (38). Hence, we have to establish an appropriate substitute for (45) where we benefit from the “minimality” of {uδ }. As in Section 2.3.1 (see in particular (24) and (25)) we modify {uδ } following Lemma B.5, i.e.: we choose for almost any ρ ◦

as above a sequence {wm } ⊂ u0 + W11 (Ω; RN ), w ˜m := wm |B2ρ (ˆx) , satisfying

38


i) wm → u∗ in L1 (Ω; RN ) as m → ∞; ii) wm |∂B2ρ (ˆx) = u∗|∂B2ρ (ˆx) ; ˜m ] = inf I = Jˆ u∗ ; B2ρ (ˆ x) ; iii) lim inf I[w m→∞

K

iv) w ˜m |Bρ (ˆx) = uδm |Bρ (ˆx) . Here {uδm } denotes a subsequence of {uδ } and analogous to Section 2.3.1 we set 1 N x); R → R , I[w] := f (∇w) dx , • I : W1 B2ρ (ˆ B (ˆ x ) 2ρ 1 N ∗ • K := w ∈ W1 B2ρ (ˆ x); R : w|∂B2ρ (ˆx) = u|∂B2ρ (ˆx) . Now, the convexity of f and Assumption 2.27, i), imply the existence of a real number ϑ > 0 such that f (Z) > ϑ if |Z| ≥ κ. Hence, we deduce from (44) and iv) x) ∩ K ϑ ≤ f (∇w ˜m ) for all x ∈ Bρ (ˆ and for all m ∈ N. This yields (recall f ≥ 0, see iii) and Proposition 2.21) |K ∩ Bρ (ˆ 1 x)| ϑ ≤ lim inf |B2ρ (ˆ x)| |B2ρ (ˆ x)| m→∞

f (∇w ˜m ) dx B2ρ (ˆ x)

1 1 inf I = x) Jˆ u∗ ; B2ρ (ˆ |B2ρ (ˆ x)| K |B2ρ (ˆ x)| 1 f (∇a u∗ ) dx = |B2ρ (ˆ x)| B2ρ (ˆx) s ∗ ∇ u 1 + d|∇s u∗ | . f∞ |B2ρ (ˆ x)| B2ρ (ˆx) |∇s u∗ | =

(46)

Both sides of (46) are independent of m and we now may pass to the limit ρ → 0. The density assumption (38) implies lim ϑ

ρ→0

|K ∩ Bρ (ˆ x)| = ϑ 2−n , |B2ρ (ˆ x)|

whereas on account of (39), (40) and the boundedness of f∞ 1 lim f (∇a u∗ ) dx ρ→0 |B2ρ (ˆ x)| B2ρ (ˆx) s ∗ ∇ u 1 + f∞ d|∇s u∗ | = 0 . |B2ρ (ˆ x)| B2ρ (ˆx) |∇s u∗ | Thus, (47) and (48) contradict (46) and Theorem 2.29 is proved.

(47)

(48)


39

2.4.2 Application: an intrinsic regularity theory for σ We finish this section with a short application of Theorem 2.29 to the regularity theory: the u∗ - and the σ-degenerate sets are identified modulo sets of measure zero and, as a consequence, an intrinsic regularity theorem for σ is obtained for the degenerate problems under consideration. To this purpose consider u∗ as given above and let a ∗ Ωdeg := u (x) = 0 x ∈ Ω : ∇ , ∗ u deg Ωσ := x ∈ Ω : σ(x) = ∇f (0) = 0 . deg The sets Ωdeg u∗ and Ωσ are well defined on the complements of sets of measure zero. For a more precise definition one has to consider Lebesgue points of σ and ∇a u∗ , respectively, where the singular part ∇s u∗ should vanish. Since ∇f is one-to-one Theorem 2.29 implies:

Corollary 2.35. With the assumptions of this section there exists a generalized minimizer u∗ ∈ M such that deg deg deg Ωu∗ − Ωdeg = Ω = 0. − Ω ∗ σ σ u On the other hand, the results of Section 2.3.1 imply: there is an open set Ωreg ⊂ Ω − Ωdeg u∗ such that for any 0 < α < 1 reg and Ω − Ωdeg − Ω u∗ ∈ C 1,α Ωreg ; RN = 0. u∗ By Theorem 2.29 (in fact (32) is sufficient), the dual solution σ is known to be of class C 0,α on Ωreg . Now observe that, again by (32) and since ∇f is one-to-one, the inclusion Ωreg ⊂ Ω − Ωdeg σ is also valid. Applying Corollary 2.35 we get the following partial regularity result for σ. Corollary 2.36. If f is given as above and if σ denotes the unique solution of the dual variational problem (P ∗ ), then there is an open set Ωreg ⊂ Ω − Ωdeg σ such that for any 0 < α < 1 σ∈C

0,α

Ω ;R reg

nN

and

deg reg Ω − Ωσ − Ω = 0 .

3 Variational integrands with (s, μ, q)-growth

In this chapter we extend the results mentioned in A and B of the table given in the introduction, i.e.: we concentrate on integrands with nearly linear and/or anisotropic growth together with some appropriate ellipticity condition. In Section 3.1 we shortly discuss a basic existence result in Orlicz-Sobolev spaces. Then we introduce the notion of (s, μ, q)-growth in Section 3.2 which is motivated by the following observation (compare also [Ma5]–[Ma7]). A theorem on the smoothness properties of solutions to our variational problem (P) is expected to depend on three free parameters: the lower growth rate s of the integrand f which provides the starting integrability of the gradient of the solution. Moreover, we need some suitable assumptions on the exponents μ and q occurring in the non-uniform ellipticity condition. We like to emphasize that this notion serves as an appropriate approach to the regularity theory of our variational problems at hand. •

The integrands discussed in the introduction are included within the notion of (s, μ, q)-growth. • The class of admissible integrands is generalized to a large extent. In particular, the known results are recovered and substantially generalized. • The variety of different settings discussed in the introduction is unified. Proving the results of this chapter, we do not have to consider several cases. • The notion of (s, μ, q)-growth is adapted to the scalar situation as well as to the vector-valued setting. Imposing the so-called (s, μ, q)-condition which relates the parameters in an appropriate way, the theorems of type (1) and (3) are proved in Section 3.3. Here we first show uniform local a priori gradient Lq -estimates for a suitable regularization. These a priori estimates are also valid in the general vectorial setting. The conclusion then follows from a DeGiorgi-type argument.


42


Partial regularity is obtained in Section 3.4 under the same hypotheses as given in Section 3.3. We benefit from the lemma on higher integrability mentioned above, and then argue using a blow-up procedure. The comparison with the known results given in Section 3.5 completes the study of the general situation. We finish this chapter by proving a theorem on the absence of singular points in the two-dimensional vector-valued case. Here, for the sake of simplicity, we restrict ourselves to energy densities with anisotropic (p, q)-growth, 1 < p ≤ q.

3.1 Existence in Orlicz-Sobolev spaces Let us start with some basic definitions: suppose that we are given a Nfunction F having the Δ2 -property. This means (see, e.g. [Ad] for details) that the function F : [0, ∞) → [0, ∞) satisfies F is continuous, strictly increasing and convex ; (N1) F (t) F (t) = 0 , lim = +∞ ; (N2) lim t↓0 t↑∞ t t there exist k, t0 ≥ 0 such that F (2t) ≤ k F (t) for all t ≥ t0 . (N3) With (N1)–(N3) F generates the Orlicz space LF (Ω; RN ) equipped with the Luxemburg norm

1 |u| dx ≤ 1 . F uLF (Ω;RN ) := inf l > 0 : l Ω We then introduce the Orlicz-Sobolev space WF1 (Ω; RN ), WF1 Ω; RN := u : Ω → RN such that u is measurable and N nN . u ∈ LF Ω; R , ∇u ∈ LF Ω; R Together with the norm uWF1 (Ω;RN ) = uLF (Ω;RN ) + ∇uLF (Ω;RnN ) this definition yields a Banach space. Moreover, we let ◦ WF1 Ω; RN = the closure of C0∞ (Ω; RN ) in WF1 Ω; RN

with respect to · WF1 (Ω;RN ) . ◦

As proved in [FO], Theorem 2.1, the space WF1(Ω; RN ) is characterized by

3.1 Existence in Orlicz-Sobolev spaces

43

Theorem 3.1. Let Ω denote a bounded Lipschitz domain. Then ◦ ◦ WF1 Ω; RN = WF1 Ω; RN ∩ W11 Ω; RN .

Throughout this chapter F is assumed to satisfy the above hypotheses. Moreover, let us introduce ◦ KF = u0 + WF1 Ω; RN ,

(1)

where the Dirichlet boundary data u0 are assumed to be of class WF1 (Ω; RN ). With these preliminaries we consider a strictly convex variational integrand f satisfying c1 F (|Z|) − c2 ≤ f (Z)

for all Z ∈ RnN ,

(2)

c1 , c2 denoting some positive numbers. Remark 3.2. Since a corresponding upper bound is not assumed in (2), we introduce in addition to (1) the energy class

1 N such that f (∇w) dx < ∞ . JF := w ∈ WF Ω; R Ω

With the above notation we have the following theorem on the existence of unique solutions in Orlicz-Sobolev classes. Theorem 3.3. If f is strictly convex and satisfies (2), then the problem f (∇w) dx → min in the class KF (P) J[w] = Ω

admits a unique solution u ∈ KF provided that we assume u0 ∈ JF . The Proof of Theorem 3.3 is given in [FO] under the additional assumption that f is bounded from above by the same N-function. In fact, this restriction is not needed since the energies of minimizing sequences are bounded by the finite value J[u0 ]. Let us sketch the main line of the proof: consider a J-minimizing sequence {um } in KF , i.e. m→∞ J[um ] −→ γ = inf J > −∞ . KF

On account of u0 ∈ JF , the growth condition (2) implies F |∇um | dx ≤ β < ∞ Ω

for some real number β > 0 which is not depending on m. We may assume that β ≥ 1 which yields by the convexity of F

44


F

1 1 |∇um | ≤ F |∇um | . β β

By the definition of · LF , we have proved that ∇um LF (Ω;RnN ) ≤ β and the Poincaré inequality (see Lemma 2.4 of [FO]) gives (3) um WF1 (Ω;RN ) ≤ c(Ω) β + u0 WF1 (Ω;RN ) . Next we recall the de la Vallèe Poussin criterion and observe that passing to a subsequence we may assume as m → ∞ um : u in W11 Ω; RN ,

◦ u ∈ u0 + W11 Ω; RN ;

um → u in L1 Ω; RN and almost everywhere in Ω . Lower semicontinuity with respect to weak W11 -convergence then implies J[u] ≤ γ. Hence, u is a suitable candidate to solve (P) and ∇u is of class LF (Ω; RnN ). It remains to establish u ∈ KF : on account of (3) we obtain as above (using in addition (N3)) F |um | dx ≤ c , Ω

where the constant is not depending on m. Moreover, almost everywhere convergence and Fatou’s lemma prove F (|u|) dx < ∞ , Ω ◦

i.e. u ∈ WF1 (Ω; RN ) and u−u0 ∈ WF1 (Ω; RN )∩ W11 (Ω; RN ). Theorem 3.1 gives u ∈ KF , hence, u is a solution of the problem (P). Note that the Δ2 -property (N3) also is needed to establish Theorem 3.1 and the Poincarè inequality of [FO]. The uniqueness of solutions is immediate by the strict convexity of f . The existence and uniqueness theorem is derived by assuming just the growthcondition (2) for the strictly convex integrand f . From now on we concentrate on the regularity theory where we need some additional conditions on the second derivatives of f . These are made precise in the next sections.

3.2 The notion of (s, μ, q)-growth – examples Definition 3.4. Let F : [0, ∞) → [0, ∞) denote some N -function satisfying (N1)–(N3). Moreover, fix some real number s ≥ 1 and assume that F (t) ≥ c0 ts for large values of t .

(4)

3.2 The notion of (s, μ, q)-growth – examples

45

A C 2 -integrand f is said to be of (s, μ, q)-growth if for all Z, Y ∈ RnN : c1 F (|Z|) − c2 ≤ f (Z) ; − μ q−2 λ 1 + |Z|2 2 |Y |2 ≤ D2 f (Z)(Y, Y ) ≤ Λ 1 + |Z|2 2 |Y |2 ,

(5) (6)

where μ ∈ R, q > 1 and c0 , c1 , c2 , λ, Λ denote positive constants. Remark 3.5. Let us briefly comment on these conditions: i) Suppose that we are given numbers q > p > 1 and that (6) holds with μ = 2−p. This case corresponds to the version of (p, q)-growth as discussed in A.2 of the introduction. ii) The logarithmic integrand and its iterated version f (Z) = |Z| ln 1 + ln 1 + · · · + ln(1 + |Z|) . . . are covered by choosing s = 1, μ = 1 (μ = 1 + ε for the iterated version) and q = 1 + ε for any ε > 0. Of course the notion of (s, μ, q)-growth also is more general than the preliminary version (3) used in B.2 of the introduction. iii) Inequality (4) together with the second part of (6) implies s ≤ q. In fact, consider the case q < 2 and observe that it is proved in [AF3], Lemma 2.1: for every γ ∈ (−1/2, 0) and μ ≥ 0 we have 1 2 γ μ + |η + s(ζ − η)|2 ds 8 2 ≤ 1 ≤ 0 γ 2γ + 1 μ + |ζ|2 + |η|2 for all ζ, η ∈ Rk , not both zero if μ = 0. Moreover, we may assume that f (0) = 0 and ∇f (0) = 0 (replace f by f − ∇f (0) : Z). This yields

1

1

D2 f (t s Z)(Z, Z) dt s ds 0 0 1 1 q−2 1 + t2 s2 |Z|2 2 dt s ds ≤ |Z|2

f (Z) =

0

0

q−2 ≤ c |Z|2 1 + |Z|2 2 , hence the upper growth rate of f is at most q (in the case q ≥ 2 this is an obvious consequence of the first inequality without referring to [AF3]). iv) Since f is smooth, convex and of maximal growth rate q, we also obtain (compare [Da], Lemma 2.2, p. 156) q−1 |∇f (Z)| ≤ c 1 + |Z|2 2 for any Z ∈ RnN and for some positive number c – in fact convexity with respect to each zαi is sufficient to prove this (see [Ma1], (2.11)).

46


v) We may assume that 2 − μ ≤ s. This is obvious if μ ≥ 1 and if μ ≤ 0, in the case 0 < μ < 1 again compare [AF3], Lemma 2.1. In the following we construct an example of an integrand satisfying (4)–(6) precisely with exponents s, μ and q for a given range of values for s, μ and q. Although the construction looks quite technical, only elementary calculations are needed. Note that Example 4.17, iii), below provides some alternative ideas. We proceed in three steps. Step 1. Let q > 1 and start with the q-power growth integrand ρ(t) = (1 + t2 )q/2 . Then we “destroy” ellipticity by defining (for all n ∈ N0 ) ρ˜(t) = ρ(t) if 2n ≤ t < 2n + 1, whereas for 2n + 1 ≤ t < 2n + 2 we let ρ˜(t) = ρ(2n + 1) + t − (2n + 1) ρ(2n + 2) − ρ(2n + 1) . Moreover, the function ρ˜ is extended to the whole line by setting ρ˜(−t) = ρ˜(t). A mollification (˜ ρ)ε with some small ε > 0 satisfies: Lemma 3.6. There is a positive constant c such that i) (˜ ρ)ε is a (smooth) N-function. ii) Let g(Z) = (˜ ρ)ε |Z| , Z ∈ RnN . Then we have for all Z, Y ∈ RnN q−2 0 ≤ D2 g(Z) Y, Y ≤ c 1 + |Z|2 2 |Y |2 . iii) The function g satisfies for any Z ∈ RnN |D2 g(Z)| |Z|2 ≤ c 1 + g(Z) . Proof of Lemma 3.6. By construction we have i). Now fix ε = 1/10 and consider the mollification with kernel k +∞ s−t −1 (˜ ρ)ε (s) = ε ρ˜(t) dt . k ε −∞ We fix n0 ∈ N and sketch ii) and iii) for a given s ∈ U (t0 ) where U (t0 ) is 0 and some small neighborhood of t0 = 2n0 + 1: to this purpose we let a = s−t ε compute ∞ k(a) lim ρ˜ (t) − lim ρ˜ (t) . (7) (˜ ρ)ε (s) = k(y) ρ (s − εy) dy + t↓t0 t↑t0 ε a Now, ρ is strictly convex which implies lim ρ˜ (t) ≤ ρ (2n + 2) and lim ρ˜ (t) ≥ ρ (2n) ,

t↓t0

t↑t0

and by (7) there is a constant (depending on ε) such that


(˜ ρ)ε (s) ≤ (ρ)ε (s) + c ρ (2n + 2) − ρ (2n) =

(ρ)ε (s)

47

(8)

+ c ρ (ξ) , ξ ∈ (2n, 2n + 2) .

Inequality (8) provides the lemma with some additional direct computations. Step 2. For the construction of a degenerate anisotropic (s, q)-power growth integrand let us assume for simplicity that n = 3 and N = 1. Suppose first that we are given numbers s ≤ q, 2 ≤ q. We let 2 2 h(Z) = (˜ ρs )ε |z1 | + (˜ ρq )ε z2 + z 3 , where (˜ ρs )ε and (˜ ρq )ε are defined as above with respect to the exponents s and q.

Fig. 1. (s, q)-growth with “linear pieces”

We now have s q c 1 + |Z|2 2 ≤ h(Z) ≤ C 1 + |Z|2 2 , q−2 0 ≤ D2 h(Z)(Y, Y ) ≤ C 1 + |Z|2 2 |Y |2

(9) (10)

48


for all Z, Y ∈ R3 with positive constants c and C. Note that the exponents in (9) and (10) cannot be improved, moreover, due to the degeneracy of D2 h, the lower bound in (10) is the best possible one. In the case 1 < q < 2 the right-hand side inequality of (10) fails to be true. Here the example is modified by letting 1 Γ := z12 + z22 + |z3 |2(1+γ) 2

h(Z) = (˜ ρs )ε (Γ) ,

for some appropriate γ > 0. Then (10) (for some q > s) and the first inequality of (9) are valid, the second one holds for some q˜, s < q˜ < q. Step 3. In the last step let us fix a μ-elliptic function Φ, i.e. Φ is assumed to satisfy the left-hand inequality of (6). Moreover, we suppose that the upper growth rate of Φ is less than or equal to s (compare Remark 3.8 and Example 3.9 for particular choices). As a result we obtain Example 3.7. Suppose that we are given numbers μ, s and q such that (2 − μ) ≤ s ≤ q . With the above notation let f (Z) = h(Z) + Φ(Z). Then f is an integrand of (s, μ, q)-growth such that the exponents cannot be improved. Remark 3.8. i) In the case 1 ≤ p := 2 − μ ≤ s ≤ q , Φ may be chosen as the power growth function Φ(Z) = (1 + |Z|2 )p/2 . ii) Non-standard ellipticity of Φ also is possible: − μ λ 1 + |Z|2 2 |Y |2 ≤ D2 Φ(Z)(Y, Y ) , μ > 1 . This will be discussed in Example 3.9. iii) If we choose f (Z) = g(Z) + Φ(Z) where g is given in Lemma 3.6, then s = q and f is “balanced” in the sense that |D2 f (Z)| |Z|2 ≤ c 1 + f (Z) (compare Lemma 3.6, iii)). We finish this section by constructing a class of non-standard μ-elliptic integrands, i.e. the ellipticity condition holds with some exponent μ > 1. Example 3.9. Given μ > 1 let

r

s

ϕ(r) = 0

− μ2

dt ds ,

r ∈ R+ 0 ,

0 |Z|

Φ(Z) = 0

1 + t2

0

s

1 + t2

− μ2

dt ds = ϕ(|Z|) , Z ∈ RnN .


49

25

20

y

15

10

5

0 -15

-10

-5

0 x

5

10

15

Fig. 2. An integrand of linear growth with μ-ellipticity, μ = 4/3.

The function Φ is of linear growth and satisfies for all Z, Y ∈ RnN with suitable constants c, C > 0 1 − μ i) DΦ(Z) = Z 1 + t2 |Z|2 2 dt ; 0 1 μ ij ∂2Φ −2 i j 2 2 −2 1 + t (Z) = δ δ − |Z| z z |Z| dt ii) αβ α β ∂zαi ∂zβj 0 − μ +|Z|−2 zαi zβj 1 + |Z|2 2 ; − μ − 1 iii) c 1 + |Z|2 2 |Y |2 ≤ D2 Φ(Z)(Y, Y ) ≤ C 1 + |Z|2 2 |Y |2 ; iv)

|D2 Φ(Z)| |Z|2 ≤ C |Z| .

∞ Proof of the above stated properties. The choice of μ shows that ϕ (r) ≤ 0 (1+ t2 )−μ/2 dt < ∞. Hence it follows that Φ is at most of linear growth. The fact that Φ is at least of linear growth is immediate by definition. Note that this in general is true for μ-elliptic integrands (compare Remark 4.2, ii), below). Using a linear transformation, the proof of i) and ii) is obvious. Moreover, the first inequality stated in iii) is a consequence of ii) and follows by considering the cases |Y : Z| ≤ 12 |Y ||Z| and |Y : Z| > 12 |Y | |Z|, respectively. We


50

like to remark that the exponent −μ/2 occurring on the left-hand side of iii) is the best possible one. This is evident if we consider Y parallel to Z. The second inequality of iii) again is immediate. Next we are going to prove iv): observing 1 − μ 2 2 1 + t2 |Z|2 2 dt |D Φ(Z)| = sup D Φ(Z)(Y, Y ) ≤ 2 |Y |=1

0

we get |D Φ(Z)| |Z| ≤ 2 |Z| 2

2

|Z|

μ 2 −2

1+s

ds ≤ 2 |Z|

0

∞

1 + s2

− μ2

ds ,

0

the last integral being finite on account of μ > 1. Remark 3.10. i) Let us already mention that Example 3.9 provides a regular class of integrands with linear growth (with some appropriate choice of μ > 1). This will be proved in the next chapter. ii) We refer to Example 4.17 for a further discussion of Φ in the case μ ≥ 1.

3.3 A priori gradient bounds and local C 1,α -estimates for scalar and structured vector-valued problems We are going to prove the Theorems 1 and 3 as stated in the introduction for our new class of variational integrals of (s, μ, q)-type. Let us assume in the vectorial setting N > 1 that f is of “special structure” in the sense that (11) f (Z) = g |Z|2 , Z ∈ RnN , holds with g: [0, ∞) → [0, ∞) of class C 2 . Note that this implies ∂2f ∂zαi ∂zβj

(Z) = 4 g |Z|2 zαi zβj + 2 g |Z|2 δ ij δαβ ,

(12)

α, β = 1, . . . , n; i, j = 1, . . . , N . We also assume in the case N > 1 that there are real numbers α ∈ (0, 1], K > 0 satisfying for all Z, Z˜ ∈ RnN 2 ˜ ≤ K |Z − Z| ˜α. D f (Z) − D2 f (Z) (13) Note that the above examples are easily adjusted to this conditions (in fact to much stronger ones) by letting

√ ε+r 2

ϕ(r) = 0

0

s

1 + t2

− μ2

dt ds ,

ε>0.

3.3 A priori gradient bounds

51

Theorem 3.11. Given μ ∈ (−∞, 2), 1 ≤ s ≤ q, 1 < q, assume that f is of (s, μ, q)-growth. In the vectorial case suppose in addition that (11) and (13) are satisfied. Finally let u denote the unique solution of the problem f (∇w) dx → min in KF (P) J[w] = Ω

with boundary values u0 ∈ JF . i) If the (s, μ, q)-condition q < (2 − μ) + s

2 n

holds, then u is of class C 1,α (Ω; RN ) for any 0 < α < 1. ii) If f is “balanced” in the sense that (compare Remark 3.8) |D2 f (Z)| |Z|2 ≤ c 1 + f (Z)

(14)

(15)

holds for all Z ∈ RnN and for some real number c > 0, then we may replace (14) by the weaker condition q < (2 − μ)

n n−2

if n > 2 .

(16)

Remark 3.12. i) Note that on account of q > 1 the condition (16) gives the upper bound μ < 1+

2 n

if n > 2 .

Our choice μ < 2 however implies that the latter inequality is valid in all dimensions. ii) Theorem 3.11 is proved in [BFM], Theorem 1.1, for the scalar case N = 1, where the assertion is formulated for local minimizers. Additionally, the results are shown to be true for (double)-obstacle problems with minimal assumptions on the obstacles. Since these aspects can be included with slight modifications of the proof given below, we keep the main line and consider unconstrained Dirichlet problems. iii) In the vectorial setting we benefit, for instance, from [FuM] and [BF3] (compare [Uh], [GiaM]). It should be mentioned that a priori gradient bounds can be derived without condition (13) which is only needed for proving local C 1,α -regularity. Proof of Theorem 3.11. The proof of Theorem 3.11 is organized in four steps: regularization, a priori Lq -estimates, a priori L∞ -estimates and the conclusion.

52


3.3.1 Regularization ˆ and boundary values In the following we fix a bounded Lipschitz domain Ω 1 ˆ N u ˆ0 ∈ Wq (Ω; R ). The data Ω and u0 of Theorem 3.11 are recovered in Sec◦

ˆ RN ) is defined as the unique tion 3.3.4. The regularization uδ ∈ u ˆ0 + Wq1 (Ω; solution of the Dirichlet problem ◦ ˆ RN , fδ (∇w) dx → min in u ˆ0 + Wq1 Ω; Jδ [w] = ˆ Ω

where for any 0 < δ < 1 q fδ (Z) := f (Z) + δ 1 + |Z|2 2 . Remark 3.13. Note that the regularization is done with respect to q. Analogous to Section 2.1.2 we could try to add a δ-term of power t > q instead of the above choice. This, however, would yield some serious technical difficulties in the scalar case (and for vectorial problems with additional structure), where it is not evident how to handle higher order δ-terms in order to make the DeGiorgi technique work. In the vector-valued setting of Section 3.4, the choice t = q gives some difficulties concerning the starting integrability since we cannot refer to the discussion of asymptotic regular integrands as given in [CE] or [GiaM]. On the other hand, once the Caccioppoli-type inequality of Lemma 3.19 is established, the absence of additional δ-terms simplifies the proof of the uniform local higher integrability Lemma 3.17. This is why we use the above approach throughout the whole Chapter. Analogous to Section 2.1.2 we have the Euler equation ◦ ˆ RN ∇fδ (∇uδ ) : ∇ϕ dx = 0 for all ϕ ∈Wq1 (Ω;

(17)

ˆ Ω

together with the uniform bound u0 ] = Jδ [uδ ] ≤ Jδ [ˆ

ˆ Ω

fδ (∇ˆ u0 ) dx .

(18)

Studying the properties of uδ let us first consider the scalar case. Then, from standard arguments (see, e.g. [LU1], Chapter 4, Theorem 5.2, p. 277) we get 2 ˆ ∩ W1 ˆ (Ω) uδ ∈ W2,loc ∞,loc (Ω) using also the fact that uδ is locally bounded which is proved under very weak assumptions in [GG]. Now fix a subdomain ˆ and a coordinate direction γ ∈ {1, . . . , n}. From (17) we obtain the Ω Ω differentiated Euler equation D2 f (∇uδ ) ∇∂γ uδ , ∇ϕ dx = 0 for all ϕ ∈ C01 Ω , Ω


53

2

f where the coefficients ∂z∂α ∂z (∇uδ ) of class C 0 (Ω ) are uniformly elliptic. β Quoting “Lp -theory” for equations with continuous coefficients (see [Mor1], 1 (Ω ) Theorem 5.5.3, p. 154, or [Gia3], Chapter 4.3, pp. 71) we get ∂γ uδ ∈ Wt,loc 2 for any finite t. Thus we have uδ ∈ Wt,loc (Ω) for any 1 ≤ t < ∞. In the vectorial case, some well known properties of uδ are summarized in the following lemma. Part I) is proved in [AF3], Proposition 2.4 and Lemma 2.5, for the second part we refer the reader to [GiaM], especially formula (3.3), and to [Ca] (compare Theorem 1.1).

Lemma 3.14. I) In the case q < 2 the approximative solution satisfies: 2 ˆ RN ; Ω; i) uδ ∈ Wq,loc 1 ˆ RnN ; Ω; ii) ∇fδ (∇uδ ) ∈ W2,loc q−2 1 ˆ RnN ; Ω; iii) 1 + |∇uδ |2 4 ∇uδ ∈ W2,loc ˆ for all M > 0. iv) ∇2 uδ 1[|∇uδ |≤M ] ∈ L2loc (Ω) II) In the case q ≥ 2 we have ˆ RN ; Ω; i) uδ ∈ W 2 2,loc

1 ˆ RnN ; Ω; ii) ∇fδ (∇uδ ) ∈ Wq/(q−1),loc q 1 ˆ iii) 1 + |∇uδ |2 4 ∈ W2,loc (Ω); q−2 1 ˆ RnN . Ω; iv) 1 + |∇uδ |2 4 ∇uδ ∈ W2,loc Remark 3.15. On account of the structure condition (11) the properties of uδ are “better” than stated in Lemma 3.14 (see, for instance, [AF3], Proposition 2.7). This information is not used for the moment, since uniform Lq -estimates do not depend on this special structure. In Section 3.3.4, Theorem 3.11 will be proved as a corollary of the uniform a priori L∞ -estimates formulated in ˆ and suppose that f satisfies Theorem 3.16. Consider a ball B2R0 (x0 ) Ω the assumptions of Theorem 3.11. Then there is a real number c = c Jδ [uδ ], R0 , depending in addition only on n, N and the data of f , such that ∇uδ L∞ (BR0 /2 (x0 );RnN ) ≤ c .


54

This Theorem will be proved in Section 3.3.3. ˆ and R0 > 0 according to the above condition In the following we fix x0 ∈ Ω ˆ Moreover, we introduce the notation c = c(f ) for constants c B2R0 (x0 ) Ω. depending only on n, N and the data of the variational integrand f . 3.3.2 A priori Lq -estimates The main part of the second step is to prove the following lemma which also holds true in the case of vectorial problems without additional structure (compare Remark 3.15). As a consequence, Lemma 3.17 will also serve as an important tool while proving the partial regularity results of the next chapter. In addition to R0 we now fix radii r, R such that 0 < r < 2R ≤ 2R0 . Lemma 3.17. Let f satisfy the assumptions of Theorem 3.11 without the restrictions (11) and (13). Moreover, let χ := n/(n − 2), if n ≥ 3. In the case n = 2 define a number χ > 1 through the condition ⎧ s ⎪ ⎨> in the case i) of Theorem 3.11; s + 2 − μ − q χ q ⎪ ⎩> in the case ii) of Theorem 3.11. 2−μ Then there are constants c ≡ c(f, r, R), β ≡ β(f ), independent of δ, such that:

(2−μ)χ 2 1 + |∇uδ |2 dx ≤ c

Br (x0 )

β 1 + fδ (∇uδ ) dx .

B2R (x0 )

Remark 3.18. Note that our assumptions imply q < (2 − μ)χ. The proof given below in fact will show that in the two-dimensional case n = 2 we can choose χ as any finite number. Of course the constants will then depend on the quantity χ. The starting point for a proof of Lemma 3.17 is the following Caccioppoli-type inequality for the approximative solutions. Lemma 3.19. There is a real number c > 0 such that for all η ∈ C01 (B2R (x0 )), 0 ≤ η ≤ 1, and for all Q ∈ RnN

η 2 D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ dx B2R (x0 ) 2 2 D fδ (∇uδ ) |∇uδ − Q|2 dx , ≤ c ∇η∞ spt∇η

where the summation with respect to γ = 1, . . . , n is always assumed. In particular, for all Q ∈ RnN


55

− μ η 2 1 + |∇uδ |2 2 |∇2 uδ |2 dx B2R (x0 ) 2 2 D fδ (∇uδ ) |∇uδ − Q|2 dx . ≤ c ∇η∞ spt∇η

Proof of Lemma 3.19. We first recall that uδ solves the regularized problem, i.e. we have (17). Next denote by eγ ∈ Rn the unit coordinate vector in xγ ˆ direction and let for a function g on Ω Δh g(x) = Δγh g(x) =

g(x + h eγ ) − g(x) , h∈R, h

denote the difference quotient of g at x in the direction eγ . Then, given Q ∈ RnN , η ∈ C01 (B2R (x0 )), the choice ϕ = Δ−h (η 2 Δh (uδ − Qx)), is admissible in (17) and an “integration by parts” implies

η 2 Δh ∇fδ (∇uδ ) : ∇Δh uδ dx B2R (x0 ) = −2 η Δh ∇fδ (∇uδ ) : Δh (uδ − Qx) ⊗ ∇η dx .

(19)

B2R (x0 )

We start with the consideration of the case q ≥ 2: by Lemma 3.14, ∇uδ is known to be of class Lrloc for some r > q and if Fh denotes the integrand on the right-hand side of (19), then the existence of a real number c(∇η), independent of h, follows such that l1 q l2 for some l1 < + |Δh uδ | , q < l2 < r. |Fh | ≤ c Δh ∇fδ (∇uδ ) q−1 Thus, equiintegrability of Fh in the sense of Vitali’s convergence theorem is established by Lemma 3.14, II), ii) and, passing to the limit h → 0, the right-hand side of (19) tends to −2 η ∂γ ∇fδ (∇uδ ) : (∂γ uδ − Qγ ) ⊗ ∇η dx ∈ (−∞, +∞) . (20) B2R (x0 )

For the left-hand side of (19) we observe

Δh ∇fδ (∇uδ ) =

1

D2 fδ ∇uδ + t h Δh ∇uδ Δh ∇uδ , · dt

0

and get using (20), Fatou’s lemma and Young’s inequality


56

η 2 D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ dx B2R (x0 ) 1 2 ≤ η lim inf D2 fδ ∇uδ + t h Δh ∇uδ Δh ∇uδ , Δh ∇uδ dt dx h→0 0 B2R (x0 ) 1 ≤ η 2 D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ dx 2 B2R (x0 ) 2 2 D fδ (∇uδ ) |∇uδ − Q|2 dx , + c ∇η∞ spt∇η

i.e. the lemma is proved for q ≥ 2. In the case q < 2 we modify the truncation arguments of [EM1]. To this purpose we fix M 1 and let for t ≥ 0 ⎧ ⎨0 , t ≥ M ψ(t) := , |ψ (t)| ≤ 4/M . ⎩ 1 , t ≤ M/2 2 Now, by Lemma 3.14, I), iv), and by [EM1], Lemma 1, ϕ = Δ η ∂γ (uδ − −h Qx) ψ(|∇uδ |) is an admissible choice, hence

η 2 ψ Δh ∇fδ (∇uδ ) : ∇∂γ uδ dx B2R (x0 ) = −2 η ψ Δh ∇fδ (∇uδ ) : ∂γ (uδ − Qx) ⊗ ∇η dx B2R (x0 ) −2 η 2 Δh ∇fδ (∇uδ ) : ∂γ (uδ − Qx) ⊗ ∇ψ dx .

(21)

B2R (x0 )

By the definition of ψ and again on account of Lemma 3.14, I), iv), both integrals on the right-hand side of (21) can be written as

Δh ∇fδ (∇uδ ) : ξ(x) dx spt η Δh ∇fδ (∇uδ ) 2 dx + ξ2 2 ≤ L (B2R (x0 );RnN ) ,

(22)

spt η

where function of class L2 . Since Lemma 3.14, I), ii), shows that ξ is a suitable ∂γ ∇fδ (∇uδ ) is of class L2loc , the strong convergence of difference quotients (see [Mor1], Theorem 3.6.8 (b), p. 84) implies (passing to the limit h → 0)

spt η

Δh ∇fδ (∇uδ ) 2 → ∂γ ∇fδ (∇uδ ) 2 a.e. , 2 Δh ∇fδ (∇uδ ) dx → ∂γ ∇fδ (∇uδ ) 2 dx . spt η

(23)


57

With (22) and (23) the variant of the Dominated Convergence Theorem given for example in [EG2], Theorem 4, p. 21, is applicable (note that the almost everywhere convergence in (23) is needed for a proof of this variant). Thus, we may pass to the limit h → 0 on the right-hand side of (21). The left-hand side is handled as in the case q ≥ 2 and summarizing the results we arrive at (again after applying Young’s inequality to the bilinear form D2 fδ (∇uδ ))

η 2 ψ D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ dx B2R (x0 ) 1 ≤ η 2 ψ D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ dx 2 B2R (x0 ) 2 2 D fδ (∇uδ ) |∇uδ − Q|2 dx + c ∇η∞ spt ∇η 2 D fδ (∇uδ ) |∇2 uδ |2 1[M/2≤|∇u |≤M ] dx . +c δ spt η

To get this inequality we note in addition that |∇u δ − Q| 0, 0 ≤ θ < 1 and A, B non-negative constants. Then there exists a constant c = c(α, θ), such that for all ρ < R, T0 ≤ ρ < R ≤ T1 we have f (ρ) ≤ c A (R − ρ)−α + B . Now let n = 2 and define α = χ(2 − μ)/2. Then we have 2 α 1 + |∇uδ | dx ≤ (η hδ )2χ dx Br (x0 )

B2R (x0 )

≤c

2χ t t ∇(η hδ ) dx ,

B2R (x0 )

where t ∈ (1, 2) is defined through 2χ = 2t/(2 − t). Using H¨ older’s inequality we get χ α 2 ∇(η hδ ) dx 1 + |∇uδ |2 dx ≤ c , Br (x0 )

B2R (x0 )

and we can proceed as before with n/(n − 2) replaced by χ. Again we have to impose the condition (1 − θ)q/(2 − μ) < 1 which for n = 2 is equivalent to χ > s/(s + 2 − μ − q). But the latter inequality follows from our choice of χ, thus Lemma 3.17 is also proved in case n = 2.


61

Remark 3.23. Once the uniform Lq -estimates are established, the weaker condition (16) is sufficient to complete the proof of both case i) and ii) as stated in Theorem 3.11. 3.3.3 Proof of Theorem 3.16 In the following we suppose that the structural condition (11) is satisfied. Moreover, we introduce the notation ωδ = ln 1 + |∇uδ |2 , A(h, r) = Aδ (h, r) = x ∈ Br (x0 ) : ωδ > h , h > 0 , ˆ and where we assume from now on that 0 < r < R < R0 . The point x0 ∈ Ω ˆ The first auxiliary the radius R0 > 0 are still fixed such that B2R0 (x0 ) Ω. result is given by Lemma 3.24. Consider η ∈ C01 (BR (x0 )), 0 ≤ η ≤ 1. Then we have for any k>0 1− μ2 1 + |∇uδ |2 |∇ωδ |2 η 2 dx A(k,R) − μ (27) + 1 + |∇uδ |2 2 (ωδ − k)2 η 2 |∇2 uδ |2 dx A(k,R) q ≤ c 1 + |∇uδ |2 2 (ωδ − k)2 |∇η|2 dx . A(k,R)

Here the constant c = c(f ) < +∞ only depends on the data and is, in particular, independent of δ and k. 2 ˆ (Ω) Proof of Lemma 3.24. i) Recall that in the scalar case uδ is of class Wt,loc for any finite t. In the vectorial setting the cases q ≥ 2 and q < 2 have to be distinguished. On account of (11) we may apply [GiaM], Theorem 3.1, to obtain in the case q ≥ 2 1 ˆ RnN ∩ L∞ ˆ nN . Ω; (28) ∇uδ ∈ W2,loc loc Ω; R

If q < 2, then the same result follows from [AF3], Proposition 2.7. As a conˆ RN ) by ∂γ ϕ, γ = 1, . . . , n, the Euler equation sequence, replacing ϕ ∈ C0∞ (Ω; (17) yields D2 fδ (∇uδ ) ∂γ ∇uδ , ∇ϕ dx = 0 (29) ˆ Ω

ˆ RN ) and for any γ = 1, . . . , n. Moreover, (28) proves that for all ϕ ∈ C0∞ (Ω; D2 fδ (∇uδ ) is of class L∞ , hence standard approximation arguments show ˆ RN ) which is compactly supported in Ω. ˆ that (29) is valid for any ϕ ∈ W21 (Ω;

62


Next we recall (see, for instance, [GT], Lemma 7.6, p. 152): given a weakly ˆ → R and writing w+ = max[w, 0] we have differentiable function w: Ω ⎧ ⎨ ∇w if w > 0 , + ∇w = ⎩0 if w ≤ 0 . Altogether, the above listed properties of ∇uδ show that ϕ = η 2 ∂γ uδ max[ωδ − k, 0] is an admissible in (29), hence D2 fδ (∇uδ ) ∂γ ∇uδ , ∇ η 2 ∂γ uδ (ωδ − k) dx = 0, for any k > 0 , A(k,R)

or equivalently D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ (ωδ − k) η 2 dx A(k,R) + D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ uδ ⊗ ∇ωδ η 2 dx A(k,R) = −2 D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ uδ ⊗ ∇η (ωδ − k) η dx .

(30)

A(k,R)

The first integral on the left-hand side is non-negative and omitted in the following estimates. Studying the second one, we first consider the scalar case and immediately obtain D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ uδ ∇ωδ η 2 dx A(k,R) 1 (31) = D2 fδ (∇uδ )(∇ωδ , ∇ωδ ) 1 + |∇uδ |2 η 2 dx 2 A(k,R) 1− μ2 ≥ c 1 + |∇uδ |2 |∇ωδ |2 η 2 dx . A(k,R)

In the vectorial setting we have to use the structure condition (11). To this ˆ → R is a weakly differentiable function, let fδ (Z) = purpose assume that ψ: Ω 2 gδ (|Z| ) and denote by ej , j = 1, . . . , N , the j th coordinate vector. Using (12) we get almost everywhere D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ uδ ⊗ ∇ψ = 4 gδ ∂α uiδ ∂γ ∂α uiδ ∂β ujδ ∂γ ujδ ∂β ψ + 2 gδ ∂γ ∂α uiδ ∂γ uiδ ∂α ψ = 2 gδ ∂γ |∇uδ |2 ∂β ψ ∂β ujδ ∂γ ujδ + gδ ∂α |∇uδ |2 ∂α ψ 1 ∂ 2 fδ (∇uδ ) ∂β ψ ∂γ |∇uδ |2 j j 2 ∂zβ ∂zγ 1 = D2 fδ (∇uδ ) ej ⊗ ∇ψ, ej ⊗ ∇|∇uδ |2 . 2 =

(32)


63

With ψ = ωδ , ∇|∇uδ |2 = (1 + |∇uδ |2 )∇ωδ and the ellipticity condition (6) of Section 3.2 we arrive at the counterpart of (31) in the case N > 1

D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ uδ ⊗ ∇ωδ η 2 dx A(k,R) 1 = D2 fδ (∇uδ ) ej ⊗ ∇ωδ , ej ⊗ ∇ωδ 1 + |∇uδ |2 η 2 dx (33) 2 A(k,R) 1− μ2 1 + |∇uδ |2 ≥ c |∇ωδ |2 η 2 dx . A(k,R)

It remains to estimate (using (32) and Young’s inequality for ε > 0) 2 2 D fδ (∇uδ ) ∂γ ∇uδ , ∂γ uδ ⊗ ∇η (ωδ − k) dx A(k,R) = D2 fδ (∇uδ ) ej ⊗ ∇η, ej ⊗ ∇ωδ (1 + |∇uδ |2 ) (ωδ − k) η dx A(k,R) ≤ ε D2 fδ (∇uδ )(ej ⊗ ∇ωδ , ej ⊗ ∇ωδ ) 1 + |∇uδ |2 η 2 dx A(k,R) 1 + D2 fδ (∇uδ )(ej ⊗ ∇η, ej ⊗ ∇η) (ωδ − k)2 1 + |∇uδ |2 dx . ε A(k,R) On account of the first equality of (33) we can absorb the ε-term if ε is sufficiently small, hence (30), (33) and (6) of Section 3.2 give

1 + |∇uδ |2

A(k,R)

≤ c A(k,R)

≤ c

1− μ2

|∇ωδ |2 η 2 dx

D2 fδ (∇uδ )(ej ⊗ ∇η, ej ⊗ ∇η) (ωδ − k)2 1 + |∇uδ |2 dx

1 + |∇uδ |2

q2

(34)

(ωδ − k)2 |∇η|2 dx .

A(k,R)

Inequality (34) shows that the first integral on the left-hand side of (27) is bounded as claimed in the lemma. 2 ii) This time we choose ϕ = η 2 ∂γ uδ max[ω − k, 0] which, as above, is seen to be admissible in (29). As in (i) we get for k > 0 D2 fδ (∇uδ )(∂γ ∇uδ , ∂γ ∇uδ ) (ωδ − k)2 η 2 dx A(k,R) +2 D2 fδ (∇uδ )(∂γ ∇uδ , ∂γ uδ ⊗ ∇ωδ ) (ωδ − k) η 2 dx A(k,R) = −2 D2 fδ (∇uδ )(∂γ ∇uδ , ∂γ uδ ⊗ ∇η) (ωδ − k)2 η dx . A(k,R)

(35)


64

Now the second integral on the left-hand side is non-negative which in the scalar case is immediate. If N > 1, then we use (32) once more to establish this observation. The right-hand side of (35) is bounded via |r.h.s.| ≤ c ε 1 + ε

D2 fδ (∇uδ )(∂γ ∇uδ , ∂γ ∇uδ ) (ωδ − k)2 η 2 dx

A(k,R)

D2 fδ (∇uδ )(∂γ uδ ⊗ ∇η, ∂γ uδ ⊗ ∇η) (ωδ − k)2 dx

,

A(k,R)

and by choosing ε > 0 properly we get from (35) D2 fδ (∇uδ )(∂γ ∇uδ , ∂γ ∇uδ ) (ωδ − k)2 η 2 dx A(k,R) ≤ c D2 fδ (∇uδ )(∂γ uδ ⊗ ∇η, ∂γ uδ ⊗ ∇η) (ωδ − k)2 dx . A(k,R)

This completes the proof of Lemma 3.24 if we recall the assumption (6), Section 3.2. Finally we introduce the notation q a(h, r) = aδ (h, r) = 1 + |∇uδ |2 2 dx , A(h,r) q τ (h, r) = τδ (h, r) = 1 + |∇uδ |2 2 (ωδ − h)2 dx , A(h,r)

to obtain Lemma 3.25. Suppose that χ > 1 is given according to Lemma 3.17. If h > k > 0, then we have for a suitable constant c = c(f ) χ−1 χ

(R − r)−2 τ (h, R) ;

i)

τ (h, r) ≤ c a(h, r)

ii)

a(h, r) ≤ (h − k)−2 τ (k, r) .

Proof of Lemma 3.25. The assertion ii) is immediate by the observation (recall the definition of A(h, r)) q q (ωδ − k)2 2 2 1 + |∇uδ | 1 + |∇uδ |2 2 dx ≤ dx . (h − k)2 A(h,r) A(h,r) ad i). We consider η ∈ C01 (BR (x0 )) such that η ≡ 1 on Br (x0 ), 0 ≤ η ≤ 1, |∇η| ≤ c (R − r)−1 . Moreover, let Γδ = Γδ (∇uδ ) = 1 + |∇uδ |2 and choose β ∈ [0, 2q ) to be fixed later. Then we have by Sobolev’s inequality in the case n ≥ 3 (recall that ∇(ωδ − h)+ = 0 on [(ωδ − h) ≤ 0])


q 2

A(h,r)

Γδ (ωδ − h) dx =

q

2

A(h,r)

Γδ2

−β

(ωδ − h)2 Γβδ dx

( q2 −β)χ

≤ A(h,r)

Γδ

χ χ−1 β

· A(h,r)

Γδ

65

χ1

(ωδ − h)2χ dx χ−1 χ

dx

!

"

=:Xδ

χ1 1 q −β 2χ ( ) ≤ Xδ (ωδ − h) dx η Γδ2 2 A(h,R) 2 1 q −β ) ( 2 2 dx , ∇ η (ωδ − h) Γ ≤ c Xδ δ

A(h,R)

and the remaining integral splits into the sum of the following terms: A(h,R)

q

|∇η|2 (ωδ − h)2 Γδ2

A(h,R)

q

η 2 |∇ωδ |2 Γδ2

−β

−β

dx

dx ≤ c (R − r)−2 τ (h, R) ; ≤ r.h.s. of inequality (27) , provided

q

q μ −β ≤ 1− ; 2 2

−β−2

η 2 (ωδ − h)2 Γδ2 |∇Γδ |2 dx A(h,R) q −β−1 ≤ c η 2 (ωδ − h)2 Γδ2 |∇2 uδ |2 dx ≤ r.h.s. of (27), A(h,R)

if again the above inequality holds for β. So let us define β = 12 (q + μ) − 1, where q ≥ 2 − μ shows β ≥ 0. Finally Xδ ≤ a(h, r)

χ−1 χ

follows from assumption (16) – of course the stronger assumption (14) also gives the result. The modifications in the case n = 2 are outlined at the end of the proof of Lemma 3.17. Here we choose χ sufficiently large such that q+μ q χ −1 ≤ . χ−1 2 2 This is possible on account of μ < 2. Altogether we have proved Lemma 3.25. Given Lemma 3.25, the well known DeGiorgi-type technique is applied – we refer to [St], Lemma 5.1, p. 219.

66


Lemma 3.26. Assume that ϕ(h, ρ) is a non-negative real valued function defined for h > k0 and ρ < R0 . Suppose further that for fixed ρ the function is non-increasing in h and that it is non-decreasing in ρ if h is fixed. Then ϕ(h, ρ) ≤

β C ϕ(k, R) , h > k > k0 , ρ < R < R0 , (h − k)α (R − ρ)γ

with some positive constants C, α, β > 1, γ, implies for all 0 < σ < 1 ϕ k0 + d, R0 − σ R0 = 0 , where the quantity d is given by β−1 2(α+β)β/(β−1) C ϕ(k0 , R0 ) d = . σ γ R0γ α

Note that Lemma 3.25 implies: for all h > k > 0, 0 < r < R < R0 we have τ (h, r) ≤ ≤

c (R − r)2 (h − k)2

χ−1 χ

c 2 χ−1 χ

(R − r)2 (h − k)

χ−1 τ (k, r) χ τ (h, R) 1+ χ−1 χ τ (k, R) .

Hence it follows from Lemma 3.26 that q τ (d, R0 /2) = 1 + |∇uδ |2 2 (ωδ − d)2 dx = 0 A(d,R0 /2)

and, as a consequence, ωδ ≤ d

on BR0 /2 (x0 ) .

(36)

Lemma 3.25 shows that d is depending on the following quantities d = d f, τ (0, R0 ), R0 . Recalling the definition τ (0, R0 ) =

1 + |∇uδ |2

BR0 (x0 )

q2

ωδ2 dx

we may write using Lemma 3.17 d = d f, Jδ [uδ ], R0 .

(37)

Finally, the definition of ωδ completes the proof of Theorem 3.16 since we have established (36) and (37).


67

3.3.4 Conclusion In the last step we are going to prove Theorem 3.11 as a corollary of Theorem 3.16. Let us suppose first that we have the additional growth estimate (38) c1 F (|Z|) − c2 ≤ f (Z) ≤ c3 1 + F (|Z|) for all Z ∈ RnN and for some real numbers c1 , c2 , c3 . Moreover, let us assume that (39) u0 ∈ Wq1 Ω; RN . ˆ = Ω. Then, at the beginning of Section 3.3.1, we may choose u ˆ 0 = u0 , Ω Furthermore, we claim that in this case uδ is a J-minimizing sequence in ◦

◦

u0 + WF1 (Ω; RN ). In fact, for any w ∈WF1 (Ω; RN ) there is a sequence {wm } in C0∞ (Ω; RN ) such that w − wm WF1 (Ω;RN ) → 0

as m → ∞ .

(40)

Now, if we have the assumption (39), then (18) gives a uniform bound for ◦

Jδ [uδ ]. Hence there is a function u ˜ ∈ u0 + WF1 (Ω; RN ) satisfying (after passing to a subsequence) as δ → 0 (41) ˜ in L1 Ω; RN , ∇uδ ∇˜ u in L1 Ω; RnN . uδ → u Lower semicontinuity and the minimality of uδ give for any ϕ ∈ C0∞ (Ω; RN ) J[˜ u] ≤ lim inf J[uδ ] ≤ lim inf Jδ [uδ ] ≤ lim sup Jδ [u0 + ϕ] δ→0

δ→0

δ→0

(42)

= J[u0 + ϕ] . We finally note that the functional J is convex and locally bounded from above on WF1 (Ω; RN ). This is a consequence of the right-hand inequality of (38). Hence J is continuous on WF1 (Ω; RN ) and the density result (40) together with (42) proves u ˜ = u, where u is the unique solution of the variational problem (P). Theorem 3.16 and (41) imply nN ) ≤ const. ∇uL∞ loc (Ω;R

(43)

Up to now (43) merely is proved if we have the additional assumptions (38) and (39). Without these assumptions we do not obtain a uniform bound for Jδ [uδ ] (recall (18)). Moreover, we cannot apply the density result (40) any longer since we lose the continuity of J on WF1 (Ω; RN ). To overcome these difficulties we consider an ε-mollification (u)ε of u and ˆ = B for some ball B Ω. Here ε > 0 is supposed to choose u ˆ0 = (u)ε and Ω be sufficiently small. Now we have uδ = uεδ . If δ = δ(ε) is given by

68


δ = δ(ε) =

1 , 1 + ε−1 + ∇(u)ε 2q q nN ) L (B,R

then the minimality of uδ implies f (∇uδ ) dx ≤ fδ (∇uδ ) dx ≤ fδ (∇(u)ε ) dx B

B

B

and from Jensen’s inequality we deduce f (∇(u)ε ) dx ≤ f (∇u) dx + O(ε) , B

B

where O(ε) → 0 as ε → 0. Next we claim √ q δ(ε) (1 + |∇(u)ε |2 ) 2 dx ≤ c(R) ε

(44)

B

with c(R) independent of ε. For the proof we observe that by the definition of δ(ε) the left-hand side of (44) is dominated by c(R) 1+x1+x 2 +ε−1 , q x := B |∇(u)ε | dx. Case 1: If x ≤ √1ε , then √ −1 √ −1 √ 1+x 1+ ε 1+ ε ε+ ε ≤ ≤ = 1 + x2 + ε−1 1 + x2 + ε−1 1 + ε−1 ε+1 √ √ ≤ ε+ ε ≤ 2 ε. Case 2: Consider the case x ≥

√1 . ε

Using 1 + x2 ≥ 12 (1 + x)2 we obtain

√ 2 1+x 2 1+x ≤ ≤ 2 ≤ ≤ ε, 1 + x2 + ε−1 1 + x2 1+x x and (44) is established. Thus, on one hand we have by (18) a uniform bound for Jδ [uδ ]. On the other hand it is proved that F ∇uδ dx ≤ f (∇uδ ) dx ≤ f (∇u) dx + O(ε) . (45) B

B

B

˜ of the sequence uδ . Lower semiHence, on B there exists a weak W11 -limit u continuity and (45) give u ˜ = u on B. As already mentioned above, Jδ [uδ ] is uniformly bounded (with our choice of δ(ε)) and Theorem 3.16 implies the claim (43) in the general setting as well. Hölder continuity of the gradient then follows by well known arguments from the Euler equation ∇f (∇u) : ∇ϕ dx = 0 for all ϕ ∈ C01 Ω; RN . Ω

3.4 Partial regularity in the general vectorial setting

69

In the scalar case N = 1 we argue with the standard difference quotient 2 (Ω). technique and, since u is Lipschitz, it follows that u is of class W2,loc Thus, letting v = ∂γ u, one arrives at D2 f (∇u) ∇v, ∇ϕ dx = 0 for all ϕ ∈ C01 Ω Ω 2

f (compare Section 3.3.1), where the coefficients ∂z∂α ∂z (∇u) are uniformly elβ older continuity of liptic on Ω Ω. Theorem 8.22, p. 200, of [GT] gives the H¨ v. In the vector-valued case an auxiliary integrand f˜ is constructed following the lines of [MS]. As a result, Theorem 3.1 of [GiaM] can be applied since we also have supposed the Hölder condition (13). Thus, the C 1,α -regularity is proved in the vectorial setting as well.

3.4 Partial regularity in the general vectorial setting The notion of (s, μ, q)-growth was introduced as an appropriate assumption to obtain local C 1,α -regularity of the unique minimizer of the variational problem (P) in the scalar case. Moreover, this result holds true in the vectorial setting with some additional structure. Going through the proof of Theorem 3.11 we already have recognized that uniform Lqloc -estimates in the sense of Lemma 3.17 do not depend on this special structure. Thus we expect that the notion of (s, μ, q)-growth is a suitable one to provide partial regularity in the general vectorial setting. In fact, we have Theorem 3.27. Let f be an integrand of (s, μ, q)-growth satisfying (14) or (15) & (16), respectively, as stated in Theorem 3.11. If u denotes the unique minimizer of the problem (P) with u0 ∈ JF , then there is an open subset Ω0 ⊂ Ω of full measure, i.e. |Ω − Ω0 | = 0, such that u ∈ C 1,α (Ω0 , RN ) for any 0 < α < 1. Here we do not assume the structure conditions (11) and (13) from Section 3.3. The Proof of Theorem 3.27 splits into four parts, the regularization, a Caccioppoli-type inequality, a blow-up lemma and a standard iteration argument. 3.4.1 Regularization Given ε > 0 we define (u)ε as the ε-mollification of u through a family of smooth mollifiers. Moreover, we fix x0 ∈ Ω and R > 0 such that BR (x0 ) ⊂ {x ∈ Ω : dist (x, ∂Ω) > ε}. Then we define the regularization uδ = uεδ as in ˆ = BR (x0 ). A comparison Section 3.3.1 where we now take u ˆ0 = (u)ε and Ω with the second case discussed in the Conclusion 3.3.4 yields

70


Lemma 3.28. If δ = δ(ε) is chosen sufficiently small and if we again write uδ = uεδ , fδ = fδ(ε) , then we have as δ → 0 (in order to use the same symbols as in the sections, where no additional parameter ε is needed, we write in the following – with a slight abuse of notation – “δ → 0”; to be more precise we should write: “ε → 0 and δ(ε) is chosen sufficiently small”) i) ii)

uδ u

δ

1 + |∇uδ |2

in W11 BR (x0 ), RN ,

q2

dx → 0 , f (∇uδ ) dx → f (∇u) dx , BR (x0 ) BR (x0 ) fδ (∇uδ ) dx → f (∇u) dx .

BR (x0 )

iii) iv)

BR (x0 )

BR (x0 )

3.4.2 A Caccioppoli-type inequality In Lemma 3.17 we already have established higher integrability of ∇u. Now we proceed with an auxiliary proposition in order to obtain the limit-version of the Caccioppoli-type inequality Lemma 3.19. Proposition 3.29. Let Θ(t) = (1 + t2 ) Θ(|∇u|). Then

2−μ 4

, t ≥ 0, hδ = Θ(|∇uδ |), h =

1 BR (x0 ) , i) h ∈ W2,loc ii) hδ h

1 BR (x0 ) as δ → 0 , in W2,loc

iii) ∇uδ → ∇u

almost everywhere on BR (x0 ) as δ → 0 .

Proof. We fix 0 < r < rˆ < R, then we combine Lemma 3.19 and Proposition 3.20 to obtain ∇hδ 2L2 (Br (x0 ),Rn ) ≤ c 1 + ∇uδ qLq (Brˆ(x0 ),RnN ) . 1 Hence, by Lemma 3.17, hδ is uniformly bounded in W2,loc (BR (x0 )) and we may assume as δ → 0 ˆ weakly in W 1 hδ : h 2,loc BR (x0 ) and almost everywhere.

To prove the pointwise convergence stated in iii), we follow [FO], Lemma 4.1, and write


BR (x0 )

71

f (∇uδ ) − f (∇u) dx

∇f (∇u) : (∇uδ − ∇u) dx 1 D2 f ∇u + t(∇uδ − ∇u)

=

BR (x0 )

+ BR (x0 )

(46)

0

(∇uδ − ∇u, ∇uδ − ∇u) (1 − t) dt dx . 1 Note that on account of u ∈ Wq,loc (Ω; RN ) and uδ ∈ Wq1 (BR (x0 ); RN ) the quantities on the right-hand side of (46) are well defined. Recall that uδ ∈ ◦

(u)ε + Wq1 (BR (x0 ); RN ), where (u)ε was a regularization of the function u, in particular we have ε→0

u − (u)ε W 1 (Ω;R ˜ N) → 0 q

Moreover,

˜ Ω. for all Ω

(47)

∇f (∇u) : (∇uδ − ∇u) dx = ∇f (∇u) : (∇uδ − ∇(u)ε ) dx BR (x0 ) + ∇f (∇u) : (∇(u)ε − ∇u) dx ;

BR (x0 )

BR (x0 )

the first term on the right-hand side is zero due to the Euler equation satisfied by u, (47) shows that second one vanishes as δ → 0 (which means ε → 0 and δ(ε) is chosen sufficiently small, compare Lemma 3.28), thus ∇f (∇u) : (∇uδ − ∇u) dx → 0 as δ → 0 . (48) BR (x0 )

In addition, by Lemma 3.28, iii), the left-hand side of (46) converges to zero and it follows from (46) that 1 δ→0 D2 f ∇u+t(∇uδ −∇u) (∇uδ −∇u, ∇uδ −∇u) (1−t) dt dx −→ 0 . BR (x0 )

0

In the case μ ≤ 0 we immediately obtain the claim, thus let us consider the case μ > 0. Then we estimate 1 D2 f ∇u + t(∇uδ − ∇u) (∇uδ − ∇u, ∇uδ − ∇u) (1 − t) dt 0 1 − μ ≥ 1 + |∇u + t(∇uδ − ∇u)|2 2 |∇uδ − ∇u|2 (1 − t) dt 0 2 − μ2 |∇uδ − ∇u|2 . ≥ c 1 + |∇u| + |∇uδ |


72

Since |∇uδ | admits pointwise almost everywhere a finite limit (we have almost everywhere convergence of hδ after passing to a subsequence), the above inequality proves that ∇uδ → ∇u almost everywhere, thus iii). As a conseˆ = h and the proof of Proposition 3.29 is complete. quence, we also have h Now we formulate the limit version of Lemma 3.19. Lemma 3.30. There is a real number c such that for all η ∈ C01 (BR (x0 )), 0 ≤ η ≤ 1, and for all Q ∈ RnN 2 2 2 2 D f (∇u) |∇u − Q|2 dx . η |∇h| dx ≤ c ∇η∞ spt ∇η

BR (x0 )

Proof. Given Q, η as above, Proposition 3.29, lower semicontinuity, Lemma 3.19 and Proposition 3.20 together imply

η |∇h| dx ≤ lim inf η 2 |∇hδ |2 dx δ→0 BR (x0 ) BR (x0 ) D2 fδ (∇uδ ) |∇uδ − Q|2 dx ≤ lim inf c ∇η2∞ δ→0 spt ∇η 2 D f (∇uδ ) |∇uδ − Q|2 dx . = lim inf c ∇η2∞ 2

δ→0

2

(49)

spt ∇η

Here, for the last equality, we made use of Lemma 3.28, ii). Next, by the pointwise convergence almost everywhere stated in Proposition 3.29, iii), we have 2 D f (∇uδ ) |∇uδ − Q|2 → D2 f (∇u) |∇u − Q|2 a.e. as δ → 0 . (50) Finally, by Lemma 3.17, we know that D2 f (∇uδ ) |∇uδ − Q|2 is uniformly bounded in L1+τ loc (BR (x0 )) for some τ > 0, hence

2 D f (∇uδ ) |∇uδ − Q|2 : ϑ 2 2 D f (∇uδ ) |∇uδ − Q| dx →

BR (x0 )

in L1+τ loc (BR (x0 )) ,

(51)

ϑ dx

BR (x0 )

as δ → 0. From (50), (51) we clearly get ϑ = |D2 f (∇u)| |∇u − Q|2 , which together with (49) completes the proof of Lemma 3.30. 3.4.3 Blow-up The next step is to prove the main decay estimate given in Lemma 3.32. Partial regularity then will follow by a standard iteration argument which is sketched in Section 3.4.4 for the readers convenience.


73

Depending on the cases q ≥ 2 and q < 2 an appropriate excess function has to be introduced: in the case q ≥ 2 we let for balls Br (x) BR (x0 ) Ω + 2 E (x, r) := − |∇u − (∇u)x,r | dy + − |∇u − (∇u)x,r |q dy , Br (x)

Br (x)

where (g)x,r denotes the mean value of the function g with respect to the ball Br (x). In the case q < 2 we define for all ξ ∈ Rk , k ∈ N, q−2 V (ξ) := 1 + |ξ|2 4 ξ . The properties of V are studied for example in [CFM], in particular we refer to Lemma 2.1 of [CFM]: Lemma 3.31. Let 1 < q < 2 and let V be defined as above. Then for any ξ, η ∈ Rk and for any t > 0 we have q−2 q q i) 2 4 min |ξ|, |ξ| 2 ≤ |V (ξ)| ≤ min |ξ|, |ξ| 2 ; ii)

q |V (t ξ)| ≤ max t, t 2 |V (ξ)|;

iii)

|V (ξ + η)| ≤ c(q) |V (ξ)| + |V (η)| ;

iv)

|V (ξ) − V (η)| q |ξ − η| ≤ q−2 ≤ c(k, q) |ξ − η|; 2 (1 + |ξ|2 + |η|2 ) 4

v) vi)

|V (ξ) − V (η)| ≤ c(k, q) |V (ξ − η)|; |V (ξ − η)| ≤ c(q, M ) |V (ξ) − V (η)| if |η| ≤ M and ξ ∈ Rk .

With these preliminaries we let for q < 2 − E (x, r) := − |V ∇u(x) − V (∇u)x,r |2 dy . Br (x)

This definition makes sense since V is of growth rate q/2 and since we have established Lemma 3.17. In both cases q ≥ 2 and q < 2 we have Lemma 3.32. Fix L > 0. Then there exists a constant C∗ (L) such that for every 0 < τ < 1/4 there is an ε = ε(L, τ ) satisfying: if Br (x) BR (x0 ) and if we have (∇u)x,r ≤ L , E(x, r) ≤ ε(L, τ ) , then E(x, τ r) ≤ C∗ (L) τ 2 E(x, r) . Here and in the following E denotes – depending on q – E + or E − , respectively.


74

Proof. The proof is organized in four steps, always distinguishing the cases q ≥ 2 and q < 2. If q ≥ 2 then we mostly refer to [FO], the case q < 2 follows the lines of [CFM] and [EM2]. 3.4.3.1 Blow-up and limit equation To argue by contradiction, assume that L > 0 is fixed, the corresponding constant C∗ (L) will be chosen later on (see Section 3.4.3.4). If Lemma 3.32 is not true, then for some 0 < τ < 1/4 there are balls Brm (xm ) BR (x0 ) such that (∇u)xm ,rm ≤ L , E(xm , rm ) =: λ2m m→∞ −→ 0 , (52) E(xm , τ rm ) > C∗ τ 2 λ2m .

(53)

Now, a sequence of rescaled functions is introduced by letting am := (u)xm ,rm , Am := (∇u)xm ,rm , um (z) :=

1 u(xm + rm z) − am − rm Am z λ m rm

if |z| ≤ 1 .

Passing to a subsequence, which is not relabeled, (52) implies Am → : A

in RnN .

(54)

We also observe that ∇um (z) = λ−1 m ∇u(xm + rm z) − Am , (um )0,1 = 0 , (∇um )0,1 = 0 , and concentrate for the moment on the case q ≥ 2. Using (52), (53) and letting Br := Br (0) we have 2 q−2 + (55) − |∇um | dz + λm − |∇um |q dz = λ−2 m E (xm , rm ) = 1 , B

B

1 1 2 q−2 − ∇um − (∇um )0,τ | dz + λm − |∇um − (∇um )0,τ |q dz > C∗ τ 2 . (56)

Bτ

Bτ

With (55) we obtain as m → ∞ um : u ˆ in W21 B1 ; RN , λm ∇um → 0 in L2 B1 ; RnN and almost everywhere , 1− 2 λm q ∇um 0 in Lq B1 ; RnN if q > 2 .

(57) (58) (59)

Considering the case q < 2 we follow [CFM], Proposition 3.4, Step 1: Lemma 3.31, ii), vi), and (54) yield


2 ∇u(x) − A m dx V λm Brm (xm ) 2 1 ≤ 2 − V ∇u(x) − Am dx λm Brm (xm ) 2 2 c (q, L) ≤ − V ∇u(x) − V (Am ) dx 2 λm

2 − V ∇um (z) dz = B1

75

−

(60)

Brm (xm )

≤ c(L) . Hence, the “q/2-growth” of V (compare Lemma 3.31, i)) implies the existence of a finite constant, independent of m, such that ∇um Lq (B1 ;RnN ) ≤ c . Thus, in the subquadratic situation (57)–(59) have to be replaced by um : u ˆ in Wq1 B1 ; RN , λm ∇um → 0 in Lq B1 ; RnN and almost everywhere .

(61) (62)

In both cases the limit u ˆ satisfies a blow-up equation stated in Proposition 3.33. There is a constant C ∗ , only depending on L, such that for all ϕ ∈ C01 (B1 ; RN ) D2 f (A)(∇ˆ u, ∇ϕ) dz = 0 , B1 2 − ∇ˆ u − (∇ˆ u)τ dz ≤ C ∗ τ 2 . (63) Bτ

Proof. The proof of the limit equation is well known (compare, for instance, [Ev], [EM2]), for the sake of completeness let us sketch the main arguments: the Euler equation for u obviously implies for any ϕ ∈ C01 (B1 ; RN ) 1 D2 f (Am + s λm ∇um )(∇um , ∇ϕ) ds dz = 0 , B1

0

i.e. we also have with ϕ fixed as above D2 f (Am )(∇um , ∇ϕ) dz B1 # 1 = − D2 f (Am + s λm ∇um )(∇um , ∇ϕ) ds B1 0 $ −D2 f (Am )(∇um , ∇ϕ) dz .

(64)

76


On account of (54) and since we have (57) and (61), respectively, the left-hand side of (64) converges to D2 f (A)(∇ˆ u, ∇ϕ) dz B1

as m → ∞. Hence, the limit-equation is proved if the right-hand side of (64) vanishes when passing to the limit m → ∞. Now, if δ > 0 is fixed, then (58) and (62), respectively, prove the existence of a measurable set S ⊂ B1 such that |B1 − S| ≤ δ and such that λm ∇um

m→∞

⇒

0

on S .

As a consequence – and with an application of H¨ older’s inequality – it only remains to show that 1 δ→0 lim D2 f (Am + s λm ∇um )(∇um , ∇ϕ) ds dz ≤ c(δ) → 0 . (65) m→∞

B1 −S

0

older’s inequality. If If |D2 f | is bounded, then (65) directly follows from H¨ −1 q > 2, then we have (considering [|∇um | ≤ (>)δ ]) q−2 1 + λ2m |∇um |2 2 |∇um | dz B1 −S

q−2 ≤ c(δ)∇um L2 (B1 ;RnN ) + δ 1−q λm |B1 − S| q−2 |∇um |q dz , +δ λm B1 −S

hence (65) and the limit equation holds on account of (55). Up to now it is proved that u ˆ is a weak solution of an elliptic system with constant coefficients. Applying the standard theory of linear elliptic systems (see [Gia1], Chapter 3), u ˆ is seen to be of class C ∞ (B1 ; RN ). We like to remark that it does not matter if q < 2: in this case we take [CFM], Proposition 2.10 as a reference. Finally, the Campanato estimate (63) of Proposition 3.33 is proved in [Gia1], Theorem 2.1 (compare also Remark 2.3). 3.4.3.2 An auxiliary proposition Proceeding in the proof of Lemma 3.32 we have to show the following proposition which will imply strong convergence in the third step. Proposition 3.34. Let q ≥ 2 and 0 < ρ < 1 or consider the case q < 2 together with 0 < ρ < 1/3. Then − μ 1 + |Am + λm ∇ˆ u + λm ∇wm |2 2 |∇wm |2 dz = 0 , lim m→∞

Bρ

ˆ. where we have set wm = um − u


77

Remark 3.35. The restriction ρ < 1/3 in the case q < 2 is needed to apply the Sobolev-Poincaré-type inequality Theorem 2.4 of [CFM]. Proof of Proposition 3.34. Again q ≥ 2 is the first case to consider, where the basic ideas are given for example in [EG1].Here we argue exactly as in [FO], pp. 410, i.e. we observe that for all ϕ ∈ C01 B1 ; RN , 0 ≤ ϕ ≤ 1,

ϕ D2 f Am + λm ∇ˆ u + s λm ∇wm (∇wm , ∇wm ) (1 − s) ds dz B1 0 −2 = λm ϕ f (Am + λm ∇um ) − f (Am + λm ∇ˆ u) dz B1 −1 − λm ϕ ∇f (Am + λm ∇ˆ u) : ∇wm dz . 1

(66)

B1

The first integral on the right-hand side of (66) can be written as B1

f (Am + λm ∇um ) dz − (1 − ϕ) f (Am + λm ∇um ) + ϕ f (Am + λm ∇ˆ u) dz , B1

where the first term can be estimated using the minimality of u, the second one using the convexity of f . As a result l.h.s of (66) ≤

λ−2 m

−λ−2 m −λ−1 m

f Am + λm ∇ um + ϕ(ˆ u − um ) dz

B1

B1

f Am + λm (1 − ϕ)∇um + ϕ∇ˆ u dz (67) ϕ ∇f (Am + λm ∇ˆ u) : ∇wm dz

B1

=: I1 − I2 − I3 . With Xm := Am + λm ((1 − ϕ)∇um + ϕ∇ˆ u) we obtain −2 f Xm + λm (ˆ u − um ) ⊗ ∇ϕ dz − f (Xm ) dz I1 − I2 = λm B1 B1 −1 ∇f (Xm ) : (ˆ u − um ) ⊗ ∇ϕ dz = λm

B1

+ B1

0

D2 f Xm + s λm (ˆ u − um ) ⊗ ∇ϕ (68) u − um ) ⊗ ∇ϕ (1 − s) ds dz . · (ˆ u − um ) ⊗ ∇ϕ, (ˆ

1

In order to derive an upper bound for the last integral, we first claim that


78

q−2 λm |um − u ˆ|q dz → 0

spt ∇ϕ

as m → ∞ .

(69)

In fact, if q = 2 then (69) immediately follows from (57). In the case q > 2 1−2/q we use (59) and Poincaré’s inequality to obtain λm um → 0 in Lq (Ω; RN ) as m → ∞. This yields together with the local gradient bound for ∇ˆ u spt ∇ϕ

λq−2 ˆ|q dz m |um − u

q1

1− q2

≤ λm →0

1− q2

u ˆLq (spt ∇ϕ;RN ) + λm

um Lq (B1 ;RN )

as m → ∞ ,

hence (69). With the notation O(m) → 0 as m → ∞ we get q−2 λm |∇um |q−2 |ˆ u − um |2 |∇ϕ|2 dz B1 q−2 q q−2 q q−2 |∇um | dz λm ≤ λm B1

|ˆ u − um |q |∇ϕ|q dz

q2

B1

= O(m) , thus starting from (68) we arrive at l.h.s of (66) ≤ c

|∇ϕ| |wm | dz + |∇ϕ| |wm | dz B1 ∇f (Xm ) : (ˆ u − um ) ⊗ ∇ϕ dz B1 ϕ ∇f (Am + λm ∇ˆ u) : ∇wm dz + O(m) . 2

q−2 λm

B1

+λ−1 m −λ−1 m

2

q

q

(70)

B1

Clearly, (70) corresponds to the inequality (6.6) in [FO]. Next we are going to discuss the last two integrals in (70), i.e.

∇f (Xm ) : (ˆ u − um ) ⊗ ∇ϕ dz − ϕ ∇f (Am + λm ∇ˆ u) : ∇wm dz B1 = ∇f (Xm ) − ∇f (Am + λm ∇ˆ u) : (ˆ u − um ) ⊗ ∇ϕ dz B1 − ∇f (Am + λm ∇ˆ u) : ∇ ϕ (um − u ˆ) dz

B1

B1

= I4 − I5 . Using 0 ≤ ϕ ≤ 1, using the upper bound for D2 f as well as the local gradient bound for u ˆ, we obtain


1

I4 = λm

79

D2 f Am + λm ∇ˆ u + s λm (1 − ϕ) (∇um − ∇ˆ u)

0

B1

u, (ˆ u − um ) ⊗ ∇ϕ (1 − ϕ) ds dz ∇um − ∇ˆ 1 2 q−2 2 1 + Am + λm ∇ˆ u + s λm (1 − ϕ) (∇um − ∇ˆ u)

≤ c λm

0

B1

|∇wm | |∇ϕ| |wm | ds dz

≤ c λm

|∇wm | |∇ϕ| |wm | dz + B1

|∇wm |

q−2 λm

q−1

|∇ϕ| |wm | dz

.

B1

For I5 we have the equality

∇f (Am + λm ∇ˆ u) − ∇f (Am ) : ∇(ϕ wm ) dz B1 1 = λm D2 f (Am + s λm ∇ˆ u) ∇ˆ u, ∇(ϕ wm ) ds dz ,

I5 =

0

B1

hence (70) implies

l.h.s of (66) ≤ c

|∇ϕ| |wm | dz + 2

2

B1

q−2 λm

|∇ϕ| |wm | dz q

q

B1

|∇wm | |∇ϕ| |wm | dz

+c B1

1

− B1

q−2 +λm

|∇wm |

q−1

|∇ϕ| |wm | dz

(71)

B1

D2 f (Am + s λm ∇ˆ u) ∇ˆ u, ∇(ϕ wm ) ds dz

0

+O(m) . Obviously the first integral on the right-hand side of (71) tends to zero as m → ∞. The second one is already discussed in (69). The third integral vanishes when passing to the limit on account of um → u ˆ in L2 (B1 ; RN ) as m → ∞. The fourth one is handled using H¨ older’s inequality and once again (69). Finally, the last integral is immediately seen to converge to zero on account of (57). Summarizing the results we have shown that

lim

m→∞

B1

1

ϕ D2 f Am + λm ∇ˆ u + s λm ∇wm (∇wm , ∇wm ) (1 − s) ds dz = 0

0

and Proposition 3.34 is proved in the case q ≥ 2. In the case q < 2 we now benefit from [EM2] (compare [Ev]) since the proof of higher integrability given in [CFM], Step 3, is adapted to balanced structure conditions. Thus, for ξ ∈ RnN we let


80

fm (ξ) :=

f (Am + λm ξ) − f (Am ) − λm ∇f (Am ) : ξ λ2m

1 and define for 0 < ρ < 1/3, w ∈ W1,loc (B1/3 , RN )

Iρm (w)

:=

Our first claim is lim sup Iρm (um ) − Iρm (ˆ u) ≤ 0 m→∞

fm (∇w) dz . Bρ

for almost every ρ ∈ (0, 1/3) .

(72)

To establish (72) we fix ρ as above, we choose 0 < s < ρ, η ∈ C0∞ (Bρ ), u − um ) η. On one 0 ≤ η ≤ 1, η ≡ 1 on Bs , |∇η| ≤ c/(ρ − s), and define ϕm = (ˆ hand, um obviously is a local minimizer of Iρm . On the other hand we have Lemma 3.3 of [CFM]: Lemma 3.36. Let f : Rk → R be a function of class C 2 satisfying for any ξ ∈ Rk q−1 |∇f (ξ)| ≤ K 1 + |ξ|2 2 , 1 < q < 2 . Then, for any M > 0 there exists a constant c depending only on M , q, K, such that if we set for any λ > 0 and A ∈ Rk with |A| ≤ M fA,λ (ξ) = λ−2 f (A + λ ξ) − f (A) − λ ∇f (A) : ξ , then

q−2 |fA,λ (ξ)| ≤ c(q, K, M ) 1 + |λ ξ|2 2 |ξ|2 .

Recalling Remark 3.5, iv), and Lemma 3.31, Lemma 3.36 yields u) ≤ Iρm (um + ϕm ) − Iρm (ˆ u) Iρm (um ) − Iρm (ˆ fm (∇um + ∇ϕm ) − fm (∇ˆ u) dz = Bρ −Bs 2 c(q, Λ, L) V (λm ∇ˆ ≤ u) 2 λm Bρ −Bs 2 dz + V λm (ˆ u − um ) ⊗ ∇η + λm η ∇ˆ u + λm (1 − η) ∇um 2 2 c V (λm ∇ˆ ≤ 2 u) + V (λm ∇um ) λm Bρ −Bs 2 +(ρ − s)−2 V λm (ˆ u − um ) dz .

(73)

Next, a family of positive, uniformly bounded Radon measures μm on B1/3 is introduced by letting


m

μ (S) := S

81

2 2 1 V (λm ∇ˆ u) + V (λm ∇um ) dz . λ2m

Repeating the computations outlined in (60) and using the smoothness of uˆ on B1/3 we obtain

2 2 1 V (λm ∇ˆ u) + V (λm ∇um ) dz μ (B1/3 ) = 2 λ B1/3 m 2 1 ≤ c 1+ V (λm ∇um ) dz ≤ c(L) , 2 λ B1/3 m m

hence we may assume that μm converges in measure to a Radon measure μ on B1/3 . Moreover, without loss of generality it is supposed in the following that (74) μ(∂Br ) = 0 for any 0 < r < 1/3 , which is true for almost any r as above. Now, the Sobolev-Poincaré type inequality stated in Theorem 2.4 of [CFM] is needed: Theorem 3.37. If 1 < q < 2 then there exist if u ∈ Wq1 (B3R (x0 ); RN ) then ⎛

2 q

< α < 2 and σ > 0 such that

1 ⎞ 2(1+σ) ⎛ 2(1+σ) ⎟ ⎜ V u − (u)x0 ,R dx⎠ ≤ c⎝ R

⎜ ⎝ − BR (x0 )

−

⎞ α1 ⎟ |V (∇u)|α dx⎠

,

B3R (x0 )

where c ≡ c(n, q, N ) is independent of R and u. To proceed further we fix σ as given in Theorem 3.37 and choose a real number 1−θ 0 < θ < 1 satisfying 12 = θ + 2 (1+σ) . Interpolation, Lemma 3.31 and Theorem 3.37 give Bρ −Bs

2 V λm (ˆ u − um ) dz

≤

Bρ −Bs

V λm (ˆ u − um ) dz

2 θ

1−θ 1+σ 2 (1+σ) V λm (ˆ · u − um ) dz Bρ −Bs 2 θ V λm (ˆ ≤ c λ2θ |ˆ u − um | dz u − um ) m

B1

Bρ

2 (1+σ) 2 (1+σ) dz u − um )0,ρ + V λm (ˆ u − um )0,ρ −λm (ˆ

1−θ 1+σ

≤


82

2 θ

≤

|ˆ u − um | dz

c λ2mθ

B1

#

2 V λm ∇(ˆ u − um ) dz

·

1+σ

1−θ $ 1+σ

2 (1+σ) + λm

B3ρ

,

and it is proved that Bρ −Bs

2 V λm (ˆ u − um ) dz ≤ c λ2m

Finally, (73) and (75) imply # u) ≤ c μm (B ρ − Bs ) + Iρm (um ) − Iρm (ˆ

2 θ

|ˆ u − um | dz

.

(75)

B1

1 (ρ − s)2

2 θ $

|ˆ u − um | dz

,

B1

hence, by taking first the limit m → ∞ and then the limit s ↑ ρ, we get (72) on account of (74). Once (72) is established for some 0 < ρ < 1/3, the following identity is the starting point to deduce a lower bound for the left-hand side: −

Iρm (um )

Iρm (ˆ u)

=

λ−2 m

u) f (Am + λm ∇um ) − f (Am + λm ∇ˆ

Bρ

−λm ∇f (Am ) : ∇wm dz 1 −1 = λm ∇f (Am + λm ∇ˆ u + s λm ∇wm ) 0

Bρ

−∇f (Am + λm ∇ˆ u) : ∇wm ds dz −1 + λm ∇f (Am + λm ∇ˆ u) − ∇f (Am ) : ∇wm dz Bρ

=: I1 + I2 . The local smoothness of u ˆ immediately implies limm→∞ I2 = 0. Since

1

1

s D2 f (Am + λm ∇ˆ u + t s λm ∇wm )(∇wm , ∇wm ) dt ds dz

I1 = Bρ

≥c

0

0

1 + |Am + λm ∇ˆ u + λm ∇wm |2

− μ2

|∇wm |2 dz

Bρ

and since we have (72), the proposition is proved for almost all, hence for all ρ ∈ (0, 1/3).


83

3.4.3.3 Strong convergence Case i): q ≥ 2 Proposition 3.38. In the case q ≥ 2 we have as m → ∞ ∇um → ∇ˆ u in L2loc B1 ; RnN ;

i) ii)

1− 2 λm q

∇um → 0

in

Lqloc B1 ; RnN

(76) if q > 2 .

Proof. Here we have to distinguish two subcases: for μ ≤ 0 the first convergence is immediate by Proposition 3.34. Using this fact, the local smoothness of u ˆ and again Proposition 3.34, the next conclusion is m→∞ 2−μ λ−μ dz −→ 0 for all 0 < ρ < 1 . (77) m |∇wm | Bρ

The proceed further, we introduce the auxiliary functions ψm (see [FO]), 2−μ 2−μ 2 2 4 4 . (78) 1 + |A ψm := λ−1 + λ ∇u | − 1 + |A | m m m m m Then, by Lemma 3.30, (57), (59) and by (6) of Section 3.2 we can estimate (0 < ρ < 1) 2 2 D f (Am + λm ∇um ) |∇um |2 dz ≤ c(ρ) . |∇ψm | dz ≤ c(ρ) Bρ

B1

If we now let Θ(Z) := (1 + |Z|2 )(2−μ)/4 , Z ∈ RnN , then |ψm | = λ−1 m

0

1

d Θ(Am + s λm ∇um ) ds ds

1 ≤ c ∇um : ∇Θ(Am + s λm ∇um ) ds 0 1 − μ ≤c |∇um | 1 + |Am + s λm ∇um |2 4 ds

0 −μ

μ

≤ c |∇um | + λm 2 |∇um |1− 2 With this inequality we obtain |ψm |2 dz ≤ c(ρ)

.

for all 0 < ρ < 1 .

(79)

Bρ

In fact, (79) is obvious for μ = 0. If μ < 0, then (79) is just a consequence of (77). Thus we have proved that

84


sup ψm W21 (Bρ ) ≤ c(ρ) < ∞

for all 0 < ρ < 1

(80)

m

and this will imply (76), ii): to this purpose we fix some real number M 1 and let Um = Um (M, ρ) := {z ∈ Bρ : λm |∇um | ≤ M }. On one hand, local L2 -convergence and q > 2 prove q−2 q q−2 q λm |∇um | dz ≤ λm |∇wm | dz + λq−2 u|q dz m |∇ˆ Um Um Um q−2 q−2 q−2 ≤ c |∇wm |2 λm |∇um | + |∇ˆ u| (81) Um q−2 + λm |∇ˆ u|q dz Um

→0

as m → ∞ .

On the other hand, observe that for M sufficiently large and for z ∈ Bρ − Um 2−μ

2 ψm (z) ≥ c λ−1 m λm |∇um (z)| μq q−2+ 2−μ

2−μ 2

,

i.e.

2q

2−μ q ψm (z) ≥ c λq−2 m |∇um (z)| .

λm

Since (14) and (16), Section 3.3, both guarantee that 2q/(2 − μ) < 2n/(n − 2), since by (80) ψm is uniformly bounded in L2n/(n−2) and since q − 2 + μq/(2 − μ) ≥ 0 follows from q ≥ 2 − μ, we can conclude q−2 λm |∇um |q dz → 0 for all 0 < ρ < 1 (82) Bρ −Um

as m → ∞. Summarizing the results, (81) and (82) prove Proposition 3.38 in the case μ ≤ 0. Now suppose that μ > 0. Proposition 3.34 implies in the case at hand (for any 0 < ρ < 1) − μ 1 + |λm ∇wm |2 2 |∇wm |2 dz → 0 as m → ∞ , Bρ

which immediately gives |∇wm |2 dz → 0

as m → ∞ .

(83)

Um

Here Um is defined as above for fixed M and ρ. Also as above we introduce ψm and observe that now |ψm | ≤ c |∇um | is obvious. Thus (80) is true in the case μ > 0 as well. If M is chosen sufficiently large, then 4

2μ

2−μ ≥ |∇um |2 |ψm | 2−μ λm

on Bρ − Um ,


85

and since 4/(2 − μ) ≤ 2n/(n − 2) ⇔ μ ≤ 4/n (where the last inequality is true on account of q ≥ 2) we get m→∞ |∇wm |2 dz −→ 0 for all 0 < ρ < 1. (84) Bρ −Um

With (83) and (84) the first claim of (76) also is proved in the case μ > 0. The second claim (76), ii) for μ > 0, follows exactly as outlined in the case μ ≤ 0. Hence, the proof of the proposition is complete. Case ii): q < 2 Proposition 3.39. If q < 2, then for any 0 < ρ < 1/3 1 V (λm ∇wm )2 dz = 0 . lim 2 m→∞ λm B ρ Proof. In the subquadratic case, the auxiliary function ψm introduced in (78) is handled via Lemma 3.31, vi). We have

|∇ψm | dz ≤ c 2

Bρ

1 + |λm ∇um |2

q−2 2

|∇um |2 dz

B

1 2 c ≤ 2 − V (λm ∇um ) dz λm B1 c V (∇u − Am )2 dx − ≤ 2 λm B(xm ,Rm ) c(L) V (∇u) − V (Am )2 dx ≤ c − ≤ 2 λm

B(xm ,Rm )

for any 0 < ρ < 1. In addition we have |ψm | ≤ c |∇um |, hence ψm ∈ 1 1 (B1 ), thus ψm ∈ Lqloc (B1 ) with q1 := nq/(n−q). Iterating this argument Wq,loc we again have sup ψm W21 (Bρ ) ≤ c(ρ) < ∞

for any 0 < ρ < 1 .

(85)

m

Assume now that 0 < ρ < 1/3. With M and Um as before, (83) once more is a consequence of Proposition 3.34. Using Lemma 3.31 we have 1 V (λm ∇wm )2 dz ≤ c V (λm ∇wm )2 dz λ2m Bρ λ2m Um c V (λm ∇um )2 dz + 2 λm Bρ −Um 2 c V (λm ∇ˆ + 2 u) dz . λm Bρ −Um

86


Then, by (83), 2 q−2 1 2 2 2 2 V (λm ∇wm ) dz ≤ 1 + λ |∇w | |∇w | dz m m m 2 λ m Um Um m→∞ ≤ |∇wm |2 dz −→ 0 . Um

The second term vanishes (passing to the limit m → ∞) provided that q−2 λm |∇um |q dz → 0 as m → ∞ . Bρ −Um

In fact, we recall the estimates for ψm stated after (81) – which remain valid in the case under consideration – and with the same reasoning we obtain (82), where now we make use of the a priori bound (85). Finally, we use the local boundedness of ∇ˆ u to get 2 1 V (λm ∇ˆ u) dz ≤ |∇ˆ u|2 dz 2 λm Bρ −Um Bρ −Um m→∞

≤ ∇ˆ u2L∞ (Bρ ,RnN ) |Bρ − Um | −→ 0 on account of λm ∇um → 0 almost everywhere on B1 as m → ∞ (see (62)). This completes the proof of Proposition 3.39. 3.4.3.4 Conclusion Proposition 3.38 and (56) yield in the case q ≥ 2 − |∇ˆ u − (∇ˆ u)τ |2 dz ≥ C∗ τ 2 , Bτ

thus we have a contradiction to (63) if we choose C∗ = 2 C ∗ . If q < 2, then we estimate according to [CFM], p. 24, E − (xm , τ rm ) 1 lim = lim m→∞ m→∞ λ2 λ2m m c ≤ lim 2 m→∞ λm

− Bτ rm (xm )

−

Bτ rm (xm )

V (∇u) − V (∇u)x

m ,τ Rm

V ∇u − (∇u)x

m ,τ Rm

2 dz

2 dz

2 c = lim 2 − V λm (∇um − (∇um )0,τ dz ≤ m→∞ λm Bτ


87

2 2 c u − (∇ˆ u)0,τ ≤ lim 2 − V (λm ∇wm ) + V λm ∇ˆ m→∞ λm Bτ 2 u)0,τ − (∇um )0,τ dz , +V λm (∇ˆ where the first integral is handled using Proposition 3.39. The last one vanishes when passing to the limit m → ∞ since we may first estimate 2 2 2 − V λm (∇ˆ u)0,τ − (∇um )0,τ dz ≤ λm − (∇ˆ u)τ − (∇um )τ dz Bτ

Bτ

and then use (61) for the right-hand side. The second integral again is estimated by (63). Thus, choosing C∗ sufficiently large we also get the contradiction in the case q < 2 and the main decay lemma is proved. 3.4.4 Iteration Finally, Theorem 3.27 follows from the iteration Lemma 3.40, which is a well known consequence of the main Lemma 3.32 (see, for instance, [GiuM1], [Ev] or [FH]). We include a short proof for the sake of completeness. Lemma 3.40. With the assumptions of Lemma 3.32 suppose that we are given numbers α ∈ (0, 1) and τ ∈ (0, 1/8) such that C∗ (2L) τ 2(1−α) ≤ 1 . Then there exits a number η = η(L, τ ) > 0 such that for every ball Br (x) BR (x0 ) the inequalities (∇u)x,r < L , (86) E(x, r) < η(L, τ ) imply for any k = 0, 1, 2, . . . (∇u)x,τ k r ≤ 2 L , E(x, τ k r) ≤ τ 2 α k E(x, r) .

(87)

(88) (89)

Proof of Lemma 3.40. The lemma is proved by induction on k. Since it is clearly true for k = 0 we now assume that (88) and (89) hold for 0 ≤ k ≤ m−1 and for some m ∈ N. With Lemma 3.32 we obtain assuming that η ≤ ε(2L, τ ) E(x, τ m r) ≤ C∗ (2L) τ 2 E(x, τ m−1 r) ≤ τ 2 α τ 2 α (m−1) E(x, r) = τ 2 α m E(x, r) ,

88


hence (89) is valid for k = m. To establish (88) with k = m we first consider the case q ≥ 2 and observe that ∇u − (∇u)x,τ k r dx (∇u)x,τ k+1 r − (∇u)x,τ k r ≤ − Bτ k+1 r (x)

1 ≤ n τ

−

∇u − (∇u)x,τ k r dx

(90)

Bτ k r (x)

≤

1 τn

+ 1 E (x, τ k r) 2 .

Thus, together with the assumptions (86) and (87) we arrive at + m−1 (∇u)x,τ m r ≤ (∇u)x,τ k+1 r − (∇u)x,τ k r + (∇u)x,r k=0

m−1 1 1 + + ≤ n E (x, τ k r) 2 + (∇u)x,r τ k=0 m−1 1 1 + αk ≤ L + n η2 τ ≤ 2L τ k=0

if η = η(L, τ ) is chosen sufficiently small. It remains to consider the case q < 2. Here we additionally observe (following [CFM]) that Lemma 3.31, i) and vi) imply for any Bρ (x) BR (x0 ) satisfying |(∇u)x,ρ | ≤ 2L − ∇u − (∇u)x,ρ dx Bρ (x)

c ≤ n ρ

Bρ (x)∩[|∇u−(∇u)x,ρ |≤1]

∇u − (∇u)x,ρ dx

∇u − (∇u)x,ρ dx

+ ρ (x)∩[|∇u−(∇u)x,ρ |>1] B c ≤ n V ∇u − (∇u)x,ρ dx ρ Bρ (x) q2 + V ∇u − (∇u)x,ρ dx Bρ (x)

12 − q1 − ≤ c(2L) E (x, ρ) + E (x, ρ) . Hence, for k ≤ m − 1 we obtain as a substitute for (90)

(91)

3.5 Comparison with some known results

89

1 1 (∇u)x,τ k+1 r − (∇u)x,τ k r ≤ c(2L) E(x, ρ) 2 + E(x, ρ) q . τn With this estimate the above arguments clearly extend to the case q < 2 and the lemma is proved. Given Lemma 3.40, Theorem 3.27 follows in the case q ≥ 2 from the definition of E and from the standard theory (compare, for example, [Gia1], Chapter 3, Theorem 1.3). If q < 2, then we have to apply (91) once again to obtain the conclusion.

3.5 Comparison with some known results It is mentioned in the introduction of this chapter that the notion of (s, μ, q)growth provides a unified and extended approach to the regularity theory for non-standard elliptic variational problems with superlinear growth. Let us shortly discuss this claim and briefly compare the main Theorems 3.11 and 3.27 with some of the basic references. We do not want to go into details and give a precise list of the assumptions supposed in the literature, we merely compare the most important hypotheses. 3.5.1 The scalar case A.2. Anisotropic power growth Consider the case N = 1 and assume that f is of (p, q)-growth as stated in (2) of the introduction. In this case, some of the most important results are due to Marcellini (see [Ma2]–[Ma4]), which – roughly speaking – read as 1 i) Assume that u ∈ Wq,loc (Ω) is a solution of the problem (P). Then u is of 1,α class C (Ω) if n q < p . n−2 1 ii) A solution u ∈ Wq,loc (Ω) exists if

q < p

n+2 . n

If we recall Theorem 3.11, then we like to remark 1 ad i). The assumption u ∈ Wq,loc (Ω) corresponds to the case s = q. Hence, the (s, μ, q)-condition (14) of Theorem 3.11 can be written as

q < p+s

2 n

⇔

q < p

n . n−2

ad ii). The existence result corresponds to the case s = p. Here we observe

90


q < p+s

2 n

⇔

q < p

n+2 . n

Thus, the above results can be interpreted as special cases of Theorem 3.11. We already mentioned that Marcellini also includes the growth rate of the energy density f in the considerations of [Ma6] and [Ma7]. On one hand, this is done in a quite general formulation. On the other hand, he has to restrict to the study of problems with superquadratic growth conditions. Nevertheless, it seems to be worth mentioning that in the case of variational problems with (s, μ, q)-growth, s ≥ 2, our (s, μ, q)-condition coincides with the set of assumptions which were introduced earlier by Marcellini from a quite different point of view. B. Growth conditions involving N-functions Again we consider the scalar case N = 1 where we now concentrate on variational integrands involving N-functions. The main contributions are 1,α -regularity of the solution in the case f (Z) = |Z| ln(1 + |Z|) (see i) Cloc [MS]). ii) In the general setting of nearly linear growth conditions the solution to the problem (P) is known to be of class C 1,α (Ω) provided that

q < 2−μ+

2 . n

This is proved in [FuM]. Clearly, these results are contained in Theorem 3.11. ad i). Choose μ = 1, s = 1, q = 1 + ε. ad ii). Consider the (s, μ, q)-condition with s = 1, i.e. q < 2−μ+s

2 n

⇔

q < 2−μ+

2 . n

3.5.2 The vectorial setting A.2. Anisotropic power growth In the vector-valued case there are only few results available for the (p, q)situation. According to Passarelli Di Napoli and Siepe (see [PS]), we have partial regularity for the solution of the problem (P) if we assume that pn 2 ≤ p < q < min p + 1, . n−1 With the choice 2 ≤ s = p, μ = 2 − p, this is a corollary of Theorem 3.27 (which, in particular, also covers the subquadratic case). However, we should emphasize the following restriction: in [PS] the parameter q is involved via

3.6 Two-dimensional anisotropic variational problems

91

c1 |Z|p ≤ f (Z) ≤ c2 1 + |Z|q . We always choose q with regard to the second derivative of f , which is a stronger assumption. This was outlined in Remark 3.5, iii). Let us finally mention the paper [AF4] which only partially fits into a comparison with Theorem 3.27 because of the very special structure taken as a hypothesis. We refer to [BF5] for a detailed discussion of examples where the structure of [AF4] leads to better results than stated above (see also Example 5.7 below). B. Growth conditions involving N-functions To our knowledge, the list of contributions on partial regularity in the case B.2 of the introduction shrinks to [FO]. In [FO] a theorem of type (2) is established in the following framework: i) q < 2; ii) μ < 4/n; iii) the integrand is balanced in some sense. Again the results are completely covered as particular cases of Theorem 3.27 (compare the variant supposing (16)). Moreover – and this seems to be even more important – the (s, μ, q)-condition enables us to handle variational problems without a balanced structure.

3.6 Two-dimensional anisotropic variational problems In Section 3.4 partial regularity results for vector-valued problems were established in the quite general setting of (s, μ, q)-growth. Now we restrict ourselves to anisotropic energy densities with (p, q)-growth. Even with this restriction and if p and q differ not too much, it seems to be unknown whether singular points can be excluded in the case of two dimensions. Here we briefly prove a theorem of this kind, to be precise we consider a bounded Lipschitz domain Ω in R2 and let f : R2N → [0, ∞) denote a function of class C 2 (R2N ) such that p−2 q−2 (92) λ 1 + |Z|2 2 |Y |2 ≤ D2 f (Z)(Y, Y ) ≤ Λ 1 + |Z|2 2 |Y |2 holds for all Z, Y ∈ R2N with positive constants λ, Λ. Here p and q are fixed exponents such that 1 < p < q < ∞. From the discussion of Remark 3.5 we obtain the growth estimate for all Z ∈ R2N (93) a |Z|p − b ≤ f (Z) ≤ A |Z|q + 1 and for suitable constants a, b, A > 0. With the help of a lemma due to Frehse and Seregin ([FrS], Lemma 4.1) we are going to prove

92


Theorem 3.41. Let f satisfy condition (92) and let u denote the solution of the problem (P), where the boundary data u0 are supposed to be of class Wp1 (Ω; RN ) such that J[u0 ] < ∞. If 1 < p < q and if we additionally have q < 2p ,

(94)

then u ∈ C 1,α (Ω; RN ) for any exponent 0 < α < 1. Remark 3.42. Note that the condition (94) formally coincides with the (s, μ, q)condition (14), Section 3.3, choosing s = p, μ = 2 − p. Example 3.43. i) As a standard model we may take q |∂1 u|2 + 1 + |∂2 u|2 2 dx Ω

for some q in the interval [2, 4). ii) Another example covered by Theorem 3.41 is given by 1+γ 2 2 2 1 + |∇u| + |∂2 u| dx Ω

with γ ∈ (0, 1]. Proof of Theorem 3.41. The proof is organized in three steps, regularization, derivation of a starting inequality and a final application of the Frehse-Seregin Lemma. Step 1. (Regularization) We refer to the regularization given in Section 3.4.1 with respect to the disc BR (0) Ω (which of course can be assumed without loss of generality), hence we have Lemma 3.28. Note that i) of Lemma 3.28 now can be replaced by (we again use the notation uδ = uεδ(ε) ). uδ u

in Wp1 BR (0); RN as δ → 0 .

(95)

In fact, i) and the growth estimate (93) immediately give (95). Now we recall the a priori Lq -estimates of Section 3.3.2. Since we have the assumption (94), Lemma 3.17 is applicable (compare Remark 3.18) and gives a real number t > q such that ∇uδ Lt (Br (0);R2N ) ≤ c(r, R) < ∞

(96)

for any radius r < R with a constant c independent of δ. In particular, u is in 1 (BR (0); RN ). the space Wt,loc Step 2. (Starting inequality) Consider a disc B2r (x0 ) ⊂ BR (0) and let η ∈ C01 (B2r (x0 )) denote a cut-off function such that η ≡ 1 on Br (x0 ), 0 ≤ η ≤ 1, and |∇η| ≤ c/r. Following the proof of Lemma 3.19 we obtain


93

η 2 D2 fδ (∇uδ )(∂γ ∇uδ , ∂γ ∇uδ ) dx BR (0) ≤ −2 η D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ [uδ − Qx] ⊗ ∇η dx

(97)

BR (0)

for any Q ∈ R2N . Next observe 2 D f (∇u ) ∇u , ∂ [u − Qx] ⊗ ∇η ∂ δ δ γ δ γ δ 12 ≤ D2 fδ (∇uδ )(∂γ ∇uδ , ∂γ ∇uδ ) 12 2 · D fδ (∇uδ ) ∂γ [uδ − Qx] ⊗ ∇η, ∂γ [uδ − Qx] ⊗ ∇η and write

Hδ =

12 . D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ

Then, uniform local higher integrability and Lemma 3.19 show that Hδ is a 2 function of class Lloc (BR (0)). From (97) we get Tr (x0 ) = B2r (x0 ) − Br (x0 ) 2 −1 Hδ dx ≤ c r Hδ |D2 fδ (∇uδ )| |∇uδ − Q| dx Br (x0 ) T (x ) 12 r 0 ≤ c r−1

Tr (x0 )

Hδ2 dx

(98)

12 2 D fδ (∇uδ ) |∇uδ − Q|2 dx .

·

Tr (x0 )

From our assumption (92) and from the definition of fδ we deduce 2 D fδ (∇uδ ) |∇uδ − Q|2 ≤ c (1 + δ) 1 + |∇uδ |2 ) q−2 2 |∇u − Q|2 . δ q−2

Let us again consider the field V (ξ) = (1 + |ξ|2 ) 4 ξ, ξ ∈ R2N – this time in the case q ≥ 2 which is assumed from now on. Then, from [Gia1], p. 151, we infer q−2 (99) 1 + |ξ|2 2 |ξ − Q|2 ≤ c |V (ξ) − V (Q)|2 , and this is exactly the place where q ≥ 2 is needed. For q < 2 the left-hand side q−2 of (99) has to be replaced by (see Lemma 3.31) (1 + |ξ|2 + |Q|2 ) 2 |ξ − Q|2 making the next calculations impossible. Now, using (99) and returning to (98), we arrive at 12 Br (x0 )

Hδ2 dx ≤ c r−1

Tr (x0 )

·

Hδ2 dx 12

|V (∇uδ ) − V (Q)|2 dx Tr (x0 )

(100)


94

with a constant c which is not depending on r, x0 and δ < 1. Since V is a diffeomorphism of R2N , we may choose Q in such a way that V (Q) = − V (∇uδ ) dx , Tr (x0 )

which enables us to estimate the second integral on the right-hand side of (100) with the help of Sobolev-Poincaré’s inequality, thus Br (x0 )

Hδ2 dx ≤ c r−1

Tr (x0 )

12 Hδ2 dx

∇ V (∇uδ ) dx .

(101)

Tr (x0 )

Note that the weak differentiability of V (∇uδ ) is already established in Lemma 3.14, II). Finally, we observe (inequality ∗ is discussed in Remark 3.44 below) q−2 ∗ ∇ V (∇uδ ) ≤ c 1 + |∇uδ |2 4 |∇2 uδ | p−2 q−p = c 1 + |∇uδ |2 4 |∇2 uδ | 1 + |∇uδ |2 4 p−2 p ≤ c 1 + |∇uδ |2 4 |∇2 uδ | 1 + |∇uδ |2 4 on account of q < 2p, and we may use the inequality (92) to get

1 + |∇uδ |2

p−2 4

|∇2 uδ | ≤ c Hδ .

p Letting hδ = 1 + |∇uδ |2 4 , (101) implies Br (x0 )

Hδ2 dx ≤ c r−1

Tr (x0 )

12 Hδ2 dx

Hδ hδ dx ,

(102)

Tr (x0 )

being valid for any disc B2r (x0 ) ⊂ BR (0). Remark 3.44. The above inequality ∗ needs a technical comment. Let us write q−2

Vδ := V (∇uδ ) = hδ p ∇uδ . In Proposition 3.20 we have proved that hδ ∈ 1 (BR (0)) and that we have W2,loc ∇hδ =

p −1 p |∇uδ | 1 + |∇uδ |2 4 ∇|∇uδ | . 2

Since we consider the two-dimensional case n = 2 and on account of hδ ≥ 1, we see that q−2 1 (BR (0)) for any t < 2, hδ p ∈ Wt,loc 2 (BR (0); RN ) implies ∇uδ ∈ Lsloc (BR (0); R2N ) for any fiand uδ ∈ W2,loc nite s. Observing also hδ ∈ Lsloc (BR (0)), s < ∞, we clearly have Vδ ∈ 1 (BR (0); R2N ) together with W1,loc


95

q−2 q−2 ∂α Vδ = ∂α hδ p ∇uδ + hδ p ∂α ∇uδ q−2 q − 2 q−2 p −1 hδ = ∂α hδ ∇uδ + hδ p ∂α ∇uδ . p

Using the formula for ∂α hδ we have proved ∗. Step 3. (Application of the Frehse-Seregin Lemma) Inequality (102) exactly corresponds to the hypotheses of Lemma 4.1 in [FrS], and we get: for any s ≥ 1 and for any compact subdomain ω of BR (0) there is a constant K = K(ω, s) such that Hδ2 dx ≤ K| ln r|−s (103) Br (x0 )

is true for any disc Br (x0 ) ω. Note that K also depends on Hδ L2 (ω) and hδ W21 (ω) but on account of Lemma 3.19, Lemma 3.28 and Proposition 3.29 with the choice μ = 2 − p, these quantities stay bounded uniformly with p−2 1 respect to δ. Let Gδ = (1 + |∇uδ |2 ) 4 ∇uδ . Recalling hδ ∈ W2,loc (BR (0)) we see that p−2 p−2 1− 2 1 + |∇uδ |2 4 = hδ p = hδ p belongs to the same function space. From Lemma 3.14 we know that uδ is 2 1 BR (0); R2N , thus Gδ ∈ W1,loc (BR (0); R2N ), and for the an element of W2,loc derivative we get (recall Proposition 3.20) |∇Gδ | ≤ c(1 + |∇uδ |2 ) that |∇Gδ |2 ≤ cHδ2 . Therefore (103) implies |∇Gδ |2 dx ≤ K | ln r|−s ,

p−2 4

|∇2 uδ | so

(104)

Br (x0 )

and if we choose s > 2 in (104), then the version of the Dirichlet-Growth Theorem given in [Fre], p.287, implies the continuity of Gδ on ω with modulus of continuity independent of δ. In Proposition 3.29, iii), we showed ∇uδ → ∇u p−2 almost everywhere on BR (0), therefore G = (1 + |∇u|2 ) 4 ∇u is a continuous p−2 function. Since ξ → (1 + |ξ|2 ) 4 ξ is a homeomorphism (note that this field p+2 is proportional to the gradient of the strictly convex potential (1 + |ξ|2 ) 4 ), we finally get continuity of ∇u. Thus, the criterion for regular points stated in Lemma 3.32 (which works in case n = 2 just under the condition (94)) is satisfied everywhere which proves the claim of Theorem 3.41 in case q ≥ 2. Let us now look at the case that (92) is valid with exponents 1 < p < q < 2. But then (92) also holds for the new choice q = 2, i.e. we have p−2 λ 1 + |Z|2 2 |Y |2 ≤ D2 f (Z)(Y, Y ) ≤ Λ|Y |2 , and we may repeat all our calculations with q replaced by the exponent 2. Since 2 < 2p, the appropriate version of condition (94) holds which gives the result.

96


Remark 3.45. i) We like to remark that it is not necessary to refer to partial regularity results, a direct proof based on the inequality (102) can be obtained as in [FrS], Theorem 2.4. ii) In [BF6] it is shown in addition that Theorem 3.41 can be extended to handle also some limit cases like f (Z) = |Z| ln(1 + |Z|) + |z2 |2 ,

Z = (z1 , z2 ) ∈ R2N ,

for which the assumption (94) is violated. By considering sufficiently regular boundary data (as done in the next chapters), we could show that minimizers are continuously differentiable functions. It remains an open question whether similar results are true for μ-elliptic variational integrands with μ > 1.

4 Variational problems with linear growth: the case of μ-elliptic integrands

In this chapter we return to variational problems with linear growth but in contrast to Chapter 2 we study the variational problem (P), ◦ (P) f (∇w) dx → min in u0 + W11 Ω; RN , J[w] := Ω

under more restrictive assumptions on the integrand f which might be sufficient for proving everywhere regularity of generalized minimizers in the scalar case N = 1 or in the vectorial case with additional structure. The hypotheses formulated below are motivated by (the linear growth) Example 3.9, in particular by the condition iii) of this example. Assumption 4.1. Throughout this chapter we consider variational integrands f satisfying the following set of hypotheses: i) f is of class C 2 (RnN ); ii) |∇f (Z)| ≤ A; − μ − 1 iii) λ 1 + |Z|2 2 |Y |2 ≤ D2 f (Z)(Y, Y ) ≤ Λ 1 + |Z|2 2 |Y |2 . Here A, λ, Λ denote positive constants, μ > 1 is some fixed exponent, and ii), iii) are valid for any choice of Z, Y ∈ RnN . An integrand f with the above properties is called μ-elliptic. Remark 4.2. Assumption 4.1 implies the following structure conditions. i) There are real numbers ν1 > 0 and ν2 such that for all Z ∈ RnN 1 ∇f (Z) : Z ≥ ν1 1 + |Z|2 2 − ν2 . ii) The integrand f is of linear growth in the sense that for real numbers ν3 > 0, ν4 , ν5 > 0, ν6 and for all Z ∈ RnN ν3 |Z| − ν4 ≤ f (Z) ≤ ν5 |Z| + ν6 .


4 Linear growth and μ-ellipticity

98

iii) The integrand satisfies a balancing condition: there is a positive number ν7 such that holds for all Z ∈ RnN . |D2 f (Z)| |Z|2 ≤ ν7 1 + f (Z) Proof. ad i). Recall that we are interested in the variational problem (P). We replace f by f¯: RnN → R, f¯(Z) := f (Z) − ∇f (0) : Z ,

Z ∈ RnN .

An integration by parts gives a real number c such that we have for all w ∈ ◦

u0 + W11 (Ω; RN )

¯ J[w] :=

f (∇w) dx −

∇f (0) : ∇w dx = J[w] + c .

Ω

Ω

¯ respecThus minimizing sequences and generalized minimizers of J and J, tively, coincide, and without loss of generality ∇f (0) = 0 can be assumed. This implies by Assumption 4.1, iii)

1

d ∇f (θ Z) : Z dθ dθ

1

D2 f (θ Z) Z, Z dθ

∇f (Z) : Z = 0

=

0

1

≥λ 0

− μ 1 + θ2 |Z|2 2 |Z|2 dθ

= λ |Z|

|Z|

1 + ρ2

− μ2

(1)

dρ ,

0

i.e. ∇f (Z) : Z is at least of linear growth and i) follows. ad ii). The upper bound is immediate by Assumption 4.1, ii). Proving the left-hand inequality we observe that (1) gives ∇f (Z) : Z ≥ 0 for all Z ∈ RnN . Without loss of generality we additionally assume that f (0) = 0 to write (using i)) 1 d f θ Z dθ f (Z) = 0 dθ 1 ≥ ∇f θ Z : θ Z dθ 1/2 $ # 1 2 2 1 |Z| ≥ − ν2 , ν1 1 + 2 4 hence ii) is clear as well. ad iii). This assertion follows from ii) and the right-hand side of Assumption 4.1, iii).


99

If μ < 1 + 2/n and if we have the structure conditions (11) and (13) of Section 3.3 in the vectorial setting, then Assumption 4.1 provides a regular class of variational integrands in the sense that generalized minimizers of the problem (P) satisfy Theorem 1 and Theorem 3 of the introduction. Moreover, the elements of the set M, i.e. generalized minimizers, merely differ by a constant. These results are established in Section 4.1. The limitation μ < 1 + 2/n was already discussed in the introduction. On the other hand, the minimal surface integrand formally is μ-elliptic with μ = 3. More precisely we have Remark 4.3. The minimal surface example f (Z) = 1 + |Z|2 satisfies Assumption 4.1 with the limit exponent μ = 3. However, there is much better information on account of the geometric structure of this example, in particular we have # $ 2 (Y · Z) c1 |Y |2 − ≤ D2 f (Z)(Y, Y ) 2 2 1 + |Z| 1 + |Z| # $ 2 c2 (Y · Z) ≤ |Y |2 − 2 1 + |Z|2 1 + |Z| for all Z, Y ∈ Rn and with some real numbers c1 , c2 . Given an integrand satisfying this condition, Ladyzhenskaya/Ural’tseva ([LU2]) and Giaquinta/Modica/Souˇcek ([GMS1]) (see also [LU3]) then use Sobolev’s inequality for functions defined on minimal hypersurfaces (compare [Mi] and [BGM]) as an essential tool for proving their regularity results. Section 4.2 is devoted to the question, whether the limit μ = 3 is of some relevance if the geometric structure condition is dropped. Here it turns out that we first have to discuss some examples. Then, with some natural boundedness assumption (which can be deduced from a suitable maximum principle), we continue by considering the vectorvalued setting: a generalized minimizer u∗ of class W11 (Ω; RN ) is found. Moreover, u∗ uniquely (up to a constant) determines the solutions of the following problem: f (∇w) dx + f∞ (u0 − w) ⊗ ν dHn−1 → min in W11 Ω; RN . (P ) Ω

∂Ω

Here f∞ denotes the recession function and ν is the unit outward normal to ∂Ω. Note that these results neither depend on a geometric structure condition nor on the assumption f (Z) = g(|Z|2 ). If – as a substitute for the geometric structure – μ < 3 is assumed, then the uniqueness of generalized BV-minimizers up to a constant as well as Theorem 1 and Theorem 3 of the introduction are also established in Section 4.2. This substantially improves the results of Section 4.1 for locally bounded generalized solutions (in the sense of Assumption 4.11 or Remark 4.12).

100


In Section 4.3 it is shown that boundedness conditions are superfluous while discussing two-dimensional problems. Here we again have to rely on the structure f (Z) = g(|Z|2 ) in the case of vector-valued problems. A complete picture is obtained by the sharpness result of Section 4.4. We benefit from ideas of [GMS1] and show that no smoothness results can be expected if the ellipticity exponent satisfies μ > 3. In view of this x-dependent example we already have included (as a model case) Section 4.2.2.2 on a smooth x-dependence.

4.1 The case μ < 1 + 2/n In this section we are going to establish our first result on full regularity and uniqueness of generalized minimizers of the problem (P), where we always assume that u0 ∈ Wp1 (Ω; RN ) for some p > 1 (by considering a suitable approximation it is also possible to handle the limit case p = 1, see Remark 2.5). Recall the notion of the set of generalized minimizers M =

u ∈ BV Ω; RN : u is the L1 -limit of a J-minimizing sequence ◦ from u0 + W11 (Ω; RN ) .

Moreover, if N > 1, we want to benefit from the above mentioned structure, thus we assume: there are constants K and 0 < α < 1 such that for all Z, Z˜ ∈ RnN (2) f (Z) = g |Z|2 , g ∈ C 2 [0, ∞); [0, ∞) , 2 ˜ ≤ K |Z − Z| ˜α. D f (Z) − D2 f (Z) (3) Now we formulate Theorem 4.4. Consider a variational integrand f satisfying Assumption 4.1 with 2 (4) μ < 1+ . n Suppose in addition that (2) and (3) are valid in the vectorial setting N > 1. Then the following assertions are true. i) Any generalized minimizer u ∈ M is in the space C 1,α (Ω; RN ) for any 0 < α < 1. ii) For u, v ∈ M we have ∇u = ∇v, i.e. generalized minimizers are unique up to a constant. Remark 4.5. In [BF3] the limit case n = 2, μ = 2 could be included with the help of John-Nirenberg estimates (see [GT], Theorem 7.21, p. 166). Here we do not follow these ideas since the technique outlined in Section 4.3 provides much stronger results.

4.1 The case μ < 1 + 2/n

101

Remark 4.6. Uniqueness up to constants as stated in Theorem 4.4, ii), is exactly what we expect recalling the known results in the minimal surface case (see the monograph of Giusti [Giu2]). The Proof of Theorem 4.4 relies on the a priori estimates given in Section 3.3. As a second ingredient the uniqueness Theorem A.9 is needed. 4.1.1 Regularization Let the assumptions of Theorem 4.4 hold. We fix some real number 1 < q < 2 satisfying q < p and in addition for n ≥ 3 (recall (4)) q < (2 − μ)

n . n−2

(5)

Now we follow exactly the lines of Section 3.3, define for any 0 < δ < 1 ◦ q 1 + |∇w|2 2 dx + J[w] , w ∈ u0 + Wq1 Ω; RN , Jδ [w] := δ Ω

and denote by uδ the unique solution of ◦ to minimize Jδ [w] in the class u0 + Wq1 Ω; RN .

q Thus, letting fδ (·) = δ 1 + | · |2 2 + f (·), we obtain ◦ ∇fδ (∇uδ ) : ∇ϕ dx = 0 for all ϕ ∈ Wq1 Ω; RN .

(Pδ )

(6)

Ω

Remark 4.7. i) It should be emphasized that this “approximative part” of the proof completely coincides with the situation of Section 3.3. ii) In particular, on account of (4) and (5), uniform local a priori gradient estimates are deduced from Theorem 3.16. Here we have to recall that the situation is a “balanced” one in the sense of Remark 4.2, iii). 4.1.2 Some remarks on the dual problem We shortly pass to the dual variational problem defined in Section 2.1.1. This will provide some auxiliary results needed in the following. The regularization at hand slightly differs from the one given in Section 2.1.2 where the case q = 2 was considered. We now let τδ := ∇f (∇uδ ) , σδ := δ Xδ + τδ = ∇fδ (∇uδ ) , q−2 Xδ := q 1 + |∇uδ |2 2 ∇uδ .

102


1 Note that σδ ∈ W2,loc (Ω; RnN ) which, for N > 1, follows from [AF3], Proposition 2.7. The arguments outlined in the second part of Section 2.1.2 now reads as: since Jδ [uδ ] ≤ Jδ [u0 ] ≤ J1 [u0 ], there exist positive numbers c1 , c2 , c3 such that q 2 2 1 + |∇uδ | δ dx ≤ c1 , f (∇uδ ) dx ≤ c2 , τδ ∞ ≤ c3 . (7) Ω

Ω

The first inequality implies , , q−1 q ,δ q Xδ , q q−1 Ω; RnN ≤ c , hence δ X 0 in L δ nN q−1 (Ω;R

L

)

(8)

as δ → 0. Here and in the following we (as usual) pass to subsequences if q necessary. By (7) and (8) we find σ ∈ L q−1 (Ω; RnN ) such that q (9) τδ , σδ : σ in L q−1 Ω; RnM as δ → 0 . Lemma 4.8. i) The limit σ given in (9) is admissible in the sense that div σ = 0. ii) Assume that σ is given as above. Then σ maximizes the dual variational problem (P ∗ ). 1 (Ω; RnN ). iii) The unique maximizer σ is of class W2,loc

iv) The sequence {uδ } is a J-minimizing sequence. Proof. The claim div σ = 0 again is a direct consequence of div σδ = 0. To prove the second assertion, recall the duality relation τδ : ∇uδ − f ∗ (τδ ) = f (∇uδ ) , which, together with the definition of σδ and with div σδ = 0, gives

2

q2

1 + |∇uδ | σδ : ∇u0 − f ∗ (τδ ) dx dx + Ω Ω q−2 1 + |∇uδ |2 2 |∇uδ |2 dx . −δ q

Jδ [uδ ] = δ

Ω

This yields for all κ ∈ L∞ (Ω; RnN ) R[κ] ≤

J[u] ≤ J[uδ ] ≤ Jδ [uδ ]

inf ◦

u∈u0 +W11 (Ω;RN )

τδ : ∇u0 − f (τδ ) dx + δ Xδ : ∇u0 dx (10) = Ω Ω q q−2 2 2 +(1 − q) δ 1 + |∇uδ | 1 + |∇uδ |2 2 dx , dx + δ q

∗

Ω

Ω

4.1 The case μ < 1 + 2/n

103

and, passing to the limit δ → 0, the second and the last integral on the righthand side both vanish according to (8) and since q < 2. As in Section 2.1.2, lower semicontinuity of Ω f ∗ (·) dx with respect to weak-∗ convergence proves the claim R[κ] ≤ R[σ] as well as q 1 + |∇uδ |2 2 dx → 0 in the limit δ → 0 . (11) δ Ω

Note that the minimizing property of {uδ } is established with (10). Finally, the arguments leading to local W21 -regularity are completely the same as given Section 2.1.3 (recall that we have the starting inequality of Lemma 3.19). However, we do not exploit iii) in the following. 4.1.3 Proof of Theorem 4.4 With Remark 4.7, ii), and Lemma 4.8, iv), it is obvious that each L1 -cluster point u∗ of the sequence {uδ } is a locally Lipschitz generalized minimizer in the class M. In particular, a generalized minimizer with vanishing singular part of the derivative is found and, following Section 2.3.1 or Appendix A.1, the Euler equation (12) ∇f (∇u∗ ) : ∇ϕ dx = 0 for all ϕ ∈ C01 Ω; RN Ω

is verified. Hence, the standard arguments as sketched at the end of Section 3.3.4 yield H¨ older continuity of the derivatives and so far it is proved that Proposition 4.9. Let the assumptions of Theorem 4.4 hold. If u∗ denotes a L1 -cluster point of the sequence {uδ }, then u∗ ∈ M and u∗ is of class C 1,α (Ω; RN ) for any 0 < α < 1. Now the arguments of Section 2.3.2 are modified to obtain Proposition 4.10. With the assumptions of Theorem 4.4 and with u∗ , σ as above we have σ = ∇f (∇u∗ ) . In particular, for any 0 < α < 1, the unique dual solution σ is of class C 0,α (Ω; RnN ). Proof. Recalling the Euler equation (12), we choose ϕ = η 2 (uδ − u∗ ), η ∈ C01 (Ω), 0 ≤ η ≤ 1. Then, together with (6), the counterpart of (29), Section 2.3.2, is established: η 2 ∇f (∇uδ ) − ∇f (∇u∗ ) : (∇uδ − ∇u∗ ) dx Ω η 2 Xδ : (∇uδ − ∇u∗ ) dx +δ Ω = −2 σδ : [uδ − u∗ ] ⊗ ∇η η dx Ω ∇f (∇u∗ ) : [uδ − u∗ ] ⊗ ∇η η dx . +2 Ω

104


Clearly the second integral on the right-hand side vanishes as δ → 0 and by (8), (11) this is also true for the second one on the left-hand side. Since the definition of σδ gives the same result for the first integral on the right-hand side, it is proved that ∇f (∇uδ ) − ∇f (∇u∗ ) : (∇uδ − ∇u∗ ) η 2 dx = 0 . lim δ↓0

Ω

Finally, the proof is completed exactly as in Section 2.3.2. Now that Proposition 4.9 and Proposition 4.10 are established, Theorem 4.4 follows as a corollary of the uniqueness result given in Theorem A.9.

4.2 Bounded generalized solutions In this section we cover the whole scale of μ-elliptic integrands with linear growth (as introduced in Assumption 4.1) up to μ = 3. As discussed in Remark 4.3, the case μ = 3 is the limit case induced by the minimal surface example, where one benefits from the geometric structure to obtain a priori gradient estimates. On account of the lack of this structure, in the general situation we have to rely on an additional assumption which, nevertheless, is a very natural one: the boundary values u0 are supposed to be of class L∞ (Ω; RN ) (following Remark 2.5, without loss of generality u0 ∈ L∞ ∩ W21 (Ω; RN )). Moreover, we suppose that a maximum principle is valid for the regularization. Note that in contrast to the previous section we again consider the standard regularization introduced in Section 2.1.2. Furthermore, the expression “bounded generalized solutions” is meant in the sense of “uniformly bounded regularizations”. Assumption 4.11. Let uδ denote the unique minimizer of ◦ δ |∇w|2 dx + J[w] , w ∈ u0 + W21 (Ω; RN ) , Jδ [w] := 2 Ω δ ∈ (0, 1). Then there is a real number M , independent of δ, such that uδ L∞ (Ω;RN ) ≤ M u0 L∞ (Ω;RN ) . Remark 4.12. i) Alternatively, Assumption 4.11 may be replaced by N ≤ K uδ L∞ loc (Ω;R )

for some real number K not depending on δ. In this case no restriction on the boundary values is needed. ii) In Section 5.1 the reader will find a formulation which seems to be quite natural for obtaining the convex hull property in the case N > 1 (compare (5) of Assumption 5.1 together with Remark 5.2, i)). Of course the convex hull property gives Assumption 4.11.

4.2 Bounded generalized solutions

105

Remark 4.13. There are a lot of contributions on the boundedness of solutions of variational problems. Let us mention [Ta] in the scalar case, a maximum principle for N > 1 is given in [DLM]. Let us also remark that in the case of non-standard growth conditions, a boundedness assumption serves as an important tool in [Ch] and [ELM2] (see Chapter 5). As outlined in the introduction, given Assumption 4.11 we do not have to differentiate the Euler equation – avoiding the use of Sobolev’s inequality – in order to obtain uniform local higher integrability of the gradients. This removes the first restriction on μ and gives an interesting existence result, even in the limit case μ = 3. Moreover, if μ < 3, then integrability is improved up to any exponent. This information finally enables us to carry out a DeGiorgi-type technique. Note that, in spite of the strong higher integrability results, this modification is non-trivial for the range of μ under consideration: for instance, let us consider the proof of Lemma 3.25 and assume that μ > 2. In this case we have to choose β > q/2 which, on the other hand, has to be excluded proving Lemma 3.25. Furthermore, observe that the conditions (14) and (16), Theorem 3.11, completely break down if μ > 2. Even the two-dimensional considerations strongly depend on the assumption μ < 2. Now let us give a precise formulation of the main results of this section. Theorem 4.14. If N ≥ 1 and if Assumption 4.1 and Assumption 4.11 are valid with μ = 3, then there exists a generalized minimizer u∗ ∈ M such that i) We have ∇s u∗ ≡ 0, hence ∇u∗ ≡ ∇a u∗ . ii) For any Ω Ω there is a constant c(Ω ) satisfying |∇u∗ | ln2 1 + |∇u∗ |2 dx ≤ c(Ω ) < ∞ . Ω

iii) The particular minimizer u∗ is of class W11 (Ω; RN ) and (up to a constant) the unique solution of the variational problem f (∇w) dx + f∞ (u0 − w) ⊗ ν dHn−1 → min in W11 Ω; RN , Ω

∂Ω

where f∞ denotes the recession function and where ν is the unit outward normal to ∂Ω. Remark 4.15. It should be emphasized that no additional structure condition is needed in the above theorem to handle the vectorial setting. A slightly stronger ellipticity condition yields: Theorem 4.16. Suppose that Assumption 4.1 and Assumption 4.11 are valid with μ < 3. In the case N > 1 we additionally assume (2) and (3) of Section 4.1. Then we have

106


i) Each generalized minimizer u ∈ M is in the space C 1,α (Ω; RN ) for any 0 < α < 1. ii) For u, v ∈ M we have ∇u = ∇v, i.e. generalized minimizers are unique up to a constant. Before we are going to prove these theorems let us briefly discuss some examples. Example 4.17. i) Consider the μ-elliptic linear growth function Φ which was introduced in Example 3.9. Suppose that N = 1, 1 < μ < 3 and fix Z, Y ∈ Rn satisfying Z = λY , λ ∈ R sufficiently large. Then # $ 2 3 (Y · Z) 1 2 −2 |Y |2 − = 1 + |Z| |Y |2 2 2 1 + |Z| 1 + |Z| − μ < 1 + |Z|2 2 |Y |2 ≤ c D2 Φ(Z)(Y, Y ) , hence Φ is not of minimal surface structure: we do not have the upper bound given in Remark 4.3. ii) In the case μ = 1, at least for large |Z|, the function Φ behaves like |Z| ln(1 + |Z|). If μ = 2, then we have the representation Φ(Z) = |Z| arctan |Z| −

1 ln 1 + |Z|2 . 2

In the limit case μ = 3 it is easy to perform the integrations with the result Φ(Z) = 1 + |Z|2 , hence the functions Φ = Φμ provide a one-parameter family connecting our logarithmic example with the minimal surface integrand. iii) On account of the above observation we need to give some examples with the limit ellipticity μ = 3, with linear growth and which are not of minimal surface structure. A quite technical example can be constructed analogous to Step 1 leading to Example 3.7: consider the function Φ for fixed 1 < μ < 3 and “destroy” the ellipticity. Then, on one hand, the upper bound being valid in the minimal surface case does not hold (see i)). On the other hand, we have degenerate ellipticity. Finally, add some minimal surface part. The reason why we need this complicated construction is the following: the Ansatz f (Z) = g(|Z|) automatically leads (more or less) to the minimal


107

surface structure. In particular, if Z ⊥ Y ∈ Rn and if f is of linear growth, hence g (t) → c as t → +∞, then − 1 g (|Z|) |Y |2 ≈ 1 + |Z|2 2 |Y |2 D f (Z)(Y, Y ) = |Z| 2

if |Z| is sufficiently large. Nevertheless, there exists a natural class of examples where the above structure is lost: the idea is to replace |Z| by the distance to a convex set. This idea provides a variety of interesting energy densities. Let us just sketch a very easy example in the case N = 1, n = 2, Z = (z1 , z2 ). Denote by C the upper unit half disc, i.e. C = {Z : |Z| < 1, z2 > 0} (for the sake of simplicity we neglect a smoothing procedure at the edges). Note that the distance function ρ(Z) := dist(Z, C) coincides (up to the constant 1) in the upper half plane (for |Z| > 1) with |Z|. Now let f (Z) = 1 + ρ2 (Z) . We are mainly interested in the points Z = (0, z2 ), z2 < 0, |z2 | 1: it evident that in this case D2 f (0, z2 ) (e1 , e1 ) = 0 , − 3 − 3 D2 f (0, z2 ) (e2 , e2 ) = 1 + |ρ2 (0, z2 ) | 2 = 1 + |Z|2 2 , where ei , i = 1, 2, denotes the ith unit coordinate vector. In particular, we observe that the minimal surface structure is completely destroyed on account of the degeneracy of C. This of course induces degeneracy of f as well. The first way of obtaining μ-elliptic integrands with linear growth evidently is to change the geometry in a suitable way. We prefer a simple and more anisotropic idea: let (for |Z| > 1) 1 τ (z1 , z2 ) = z1 + ρ (z1 , z2 ) , 2 f˜ (z1 , z2 ) = 1 + τ 2 (z1 , z2 ) . Then there is a positive constant c such that c−1 |Z| ≤ τ (z1 , z2 ) ≤ c |Z| for all |Z| sufficiently large, and if z2 < 0, |z2 | 1, then we obtain in both coordinate directions ei , i = 1, 2, with suitable constants ci − 3 D2 f˜ (0, z2 ) (ei , ei ) = ci 1 + |τ (0, z2 ) |2 2 . Summarizing the properties of f˜ we see that this function is of linear growth and satisfies the μ-ellipticity condition with the limit exponent μ =

108


3. Moreover, f˜ does not satisfy the minimal surface ellipticity condition from Remark 4.3 and there is no chance to get something analogous: given the points (0, z2 ) as above, both eigenvalues of D2 f˜ (0, z2 ) grow like (1 + |Z|2 )−3/2 . 4.2.1 Regularization In contrast to the previous chapter and in contrast to Section 4.1.1, no particular choice of the power of the regularization is induced by the variational problem, hence we consider the regularization as given in Assumption 4.11. Once again note that we have (letting fδ := 2δ | · |2 + f ) the uniform bound fδ (∇uδ ) dx ≤ c (13) Ω

for some real number c as well as the Euler equation ◦ ∇fδ (∇uδ ) : ∇ϕ dx = 0 for all ϕ ∈W21 Ω; RN .

(14)

Ω

As in Section 2.1.2 we denote σδ = ∇fδ (∇uδ ) and assume on account of (13) that σδ : σ in L2 (Ω; RnN ) as δ → 0. Remark 4.18. Recall that the following claims were proved in Section 2.1.2. i) The sequence {uδ } is a J-minimizing sequence. Hence, the L1 -cluster points of {uδ } provide generalized minimizers u∗ ∈ M. ii) The limit σ of the sequence {σδ } maximizes the dual variational problem (P ∗ ) which was introduced in Section 2.1.1. Next, Lemma 3.19 and Lemma 3.24 have to be generalized. Lemma 4.19. Suppose that Assumption 4.1 is true and that we have the structure condition (2), Section 4.1, in the case N > 1. i) There is a real number c > 0 such that for any s ≥ 0, for all η ∈ C0∞ (Ω), 0 ≤ η ≤ 1 and for any δ ∈ (0, 1) D2 fδ (∇uδ )(∂γ ∇uδ , ∂γ ∇uδ ) Γsδ η 2 dx Ω D2 fδ (∇uδ )(∂γ uδ ⊗ ∇η, ∂γ uδ ⊗ ∇η) Γsδ dx , ≤ c Ω

where we have set Γδ = 1 + |∇uδ |2 . ii) In contrast to Section 3.3.3 we denote in the following A(k, r) = Aδ (k, r) = x ∈ Br (x0 ) : Γδ > k ,

k>0.


109

Then there is a real number c > 0, independent of δ, such that for all η ∈ C0∞ (Br (x0 )), 0 ≤ η ≤ 1 and for any δ ∈ (0, 1)

−μ

Γδ 2 |∇Γδ |2 η 2 dx A(k,r) ≤ c D2 fδ (∇uδ )(ej ⊗ ∇η, ej ⊗ ∇η) (Γδ − k)2 dx . A(k,r)

Here ej denotes the j th coordinate vector. Remark 4.20. Following the proof of Lemma 4.19 we see that the structure condition is not needed to establish the first claim in the case s = 0 (compare Lemma 3.19). Proof of Lemma 4.19. ad i). We have already seen at the beginning of Section 2.1.3 that the Euler equation (14) yields (15) D2 fδ (∇uδ )(∂γ ∇uδ , ∇ϕ) dx = 0 for all ϕ ∈ C0∞ Ω; RN , Ω

hence, using standard approximation arguments, (15) follows for all ϕ ∈ W21 (Ω; RN ) which are compactly supported in Ω. Next we cite [LU1], Chapter 4, Theorem 5.2, in the scalar case and [Uh] (we may also refer to [GiaM], 1 Theorem 3.1) if N > 1 to see that uδ ∈ W∞,loc (Ω; RN ). As a consequence, ϕ = η 2 ∂γ uδ Γsδ (with η given above) is admissible in (15) (recall the product and the chain rule for Sobolev functions). Summarizing the results we arrive at D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ Γsδ η 2 dx Ω (16) +s D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ uδ ⊗ ∇Γδ Γδs−1 η 2 dx Ω = −2 D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ uδ ⊗ ∇η η Γsδ dx . Ω

In the scalar case N = 1 the second integral on the left-hand side can be neglected on account of 1 D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ uδ ⊗ ∇Γδ = D2 fδ (∇uδ ) ∇Γδ , ∇Γδ ≥ 0 a.e. 2 In the vectorial setting N > 1 we first consider the case s = 0. Then the second term on the left-hand side trivially vanishes without any additional assumption (compare Remark 4.20). If s > 0, then the structure condition is needed: given a weakly differentiable function ψ: Ω → R and letting fδ (Z) = gδ (|Z|2 ) we have proved in (32) of Section 3.3.3 that almost everywhere


110

D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ uδ ⊗ ∇ψ 1 = D2 fδ (∇uδ ) ej ⊗ ∇ψ, ej ⊗ ∇Γδ . 2

(17)

Choosing ψ = Γδ we see that in the vectorial setting the second integral on the left-hand side of (16) is non-negative as well. In any case we obtain for a given ε > 0 D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ Γsδ η 2 dx Ω s 12 2 ≤ c η Γδ2 D fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ Ω 12 2s 2 · D fδ (∇uδ ) ∂γ uδ ⊗ ∇η, ∂γ uδ ⊗ ∇η Γδ dx ≤ c ε D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ Γsδ η 2 dx Ω

s −1 2 +ε D fδ (∇uδ ) ∂γ uδ ⊗ ∇η, ∂γ uδ ⊗ ∇η Γδ dx . Ω

If ε is sufficiently small, then we may absorb the first integral on the right-hand side and i) is proved. ad ii). This time we choose ϕ = η 2 ∂γ uδ max Γδ − k, 0 . The arguments given in i) again show that this provides an admissible choice for (15):

D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ (Γδ − k) η 2 dx A(k,r) + D2 fδ (∇uδ )(∂γ ∇uδ , ∂γ uδ ⊗ ∇Γδ η 2 dx A(k,r) = −2 D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ uδ ⊗ ∇η η (Γδ − k) dx .

(18)

A(k,r)

Here the non-negative first integral on the left-hand side is neglected and the second integral is estimated as above: 1 D2 fδ (∇uδ ) ej ⊗ ∇Γδ , ej ⊗ ∇Γδ η 2 dx 2 A(k,r) (19) ≤ D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ uδ ⊗ ∇Γδ η 2 dx . A(k,r)

According to (17), the right-hand side of (18) satisfies almost everywhere 1 D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ uδ ⊗ ∇η = D2 fδ (∇uδ ) ej ⊗ ∇η, ej ⊗ ∇Γδ . (20) 2 Inequalities (18)–(20) imply with the Cauchy-Schwarz inequality


111

D2 fδ (∇uδ ) ej ⊗ ∇Γδ , ej ⊗ ∇Γδ η 2 dx A(k,r) ≤ c ε D2 fδ (∇uδ ) ej ⊗ ∇Γδ , ej ⊗ ∇Γδ η 2 dx A(k,r)

+ε−1

D2 fδ (∇uδ ) ej ⊗ ∇η, ej ⊗ ∇η (Γδ − k)2 dx ,

A(k,r)

hence ii) is proved by recalling Assumption 4.1, iii), and by choosing ε > 0 sufficiently small. 4.2.2 The limit case μ = 3 4.2.2.1 Higher local integrability Now we concentrate on the limit case μ = 3, prove local uniform integrability of |∇uδ | ln2 (1 + |∇uδ |2 ) and establish the first two parts of Theorem 4.14 for any weak cluster point u∗ of {uδ }. The last assertion then is a corollary of Theorem A.11, hence, we have a Proof of Theorem 4.14. Note that the discussion of the vectorial setting does not depend on additional conditions (compare Remark 4.15). Theorem 4.21. Let Assumption 4.1 and Assumption 4.11 hold in the limit case μ = 3. Then for any Ω Ω there is a real number c(Ω ) – independent of δ – such that |∇uδ | ln2 1 + |∇uδ |2 dx ≤ c(Ω ) < ∞ . Ω

Proof. Given B2r (x0 ) Ω we first have to show that ϕ = uδ ωδ2 η 2 , ωδ = ln(Γδ ), η ∈ C0∞ (B2r (x0 )), 0 ≤ η ≤ 1, η ≡ 1 on Br (x0 ), is admissible in the Euler equation (14). Since the vectorial structure condition is dropped in this section, we refer to the discussion of asymptotically regular integrands given in [CE]; a generalization is proved in [GiaM], Theorem 5.1. As a result, uδ is 2 1 ∩W∞,loc (Ω; RN ) which proves that ϕ is an admissible seen to be of class W2,loc choice. Alternatively, we could replace ωδ by a suitable truncation ωδ,M and prove Theorem 4.21 by passing to the limit M → ∞. With the above choice, the Euler equation reads as

∇uδ ωδ2 η 2

∇f (∇uδ ) : dx + δ |∇uδ |2 ωδ2 η 2 dx B2r (x0 ) B2r (x0 ) 2 2 = − ∇f (∇uδ ) : uδ ⊗ ∇ωδ η + ∇η 2 ωδ2 dx B2r (x0 ) −δ ∇uδ : uδ ⊗ ∇ωδ2 η 2 + ∇η 2 ωδ2 dx . B2r (x0 )

(21)


112

Remark 4.2, i), proves that the left-hand side of (21) is greater than or equal to 1 2 2 2 2 2 ν1 Γδ ωδ η − ν2 ωδ η dx + δ |∇uδ |2 ωδ2 η 2 dx . (22) B2r (x0 )

B2r (x0 )

Since |∇f | and |uδ | are bounded, we find an upper bound for the right-hand side of (21) (using Young’s inequality with ε > 0 fixed) r.h.s ≤ c

η

2

B2r (x0 )

+c(r) +c δ

1 2

ε Γδ ωδ2

B2r (x0 )

+ε

−1

−1 Γδ 2

|∇ωδ |

2

dx

ωδ2 dx

η ε |∇uδ | 2

2

B2r (x0 )

+c(r) δ B2r (x0 )

ωδ2

+ε

−1

|∇ωδ |

2

(23)

dx

|∇uδ | ωδ2 dx .

Clearly B2r (x0 ) ωδ2 dx and δ B2r (x0 ) |∇uδ |ωδ2 dx are uniformly bounded with respect to δ (compare (13)). Hence, (21)–(23) imply after absorbing terms (for ε sufficiently small) Br (x0 )

1 2

Γδ ωδ2

−1 dx ≤ c 1 + Γδ 2 |∇ωδ |2 η 2 dx . B2r (x0 ) 2 2 +δ |∇ωδ | η dx .

(24)

B2r (x0 )

Given (24) we observe that almost everywhere |∇ωδ |2 ≤ c

1 |∇2 uδ |2 , 1 + |∇uδ |2

thus we may use Assumption 4.1, iii), with μ = 3, Lemma 4.19 (letting s = 0 and recalling Remark 4.20) as well as Remark 4.2, iii), and (13) to obtain the final result . 1 1 −2 −1 2 2 2 2 2 Γδ + δ Γδ |∇ uδ | η dx Γδ ωδ dx ≤ c 1 + c Br (x0 ) B2r (x0 ) . 2 2 ≤ c 1+c D fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ η dx B2r (x0 ) . 2 2 D fδ (∇uδ ) |∇uδ | dx ≤ c . ≤ c 1 + c(r) B2r (x0 )


113

4.2.2.2 The independent variable Throughout our whole studies we consider autonomous energy densities f , the case f = f (x, P ), x ∈ Ω, P ∈ RnN is omitted for the sake of technical simplicity. Here we take variational problems with linear growth and with limit ellipticity as a model to show that a smooth x-dependence does not affect our results. Of course, a sufficiently smooth dependence on x can be considered in the superlinear case as well. We have chosen the linear setting with very weak ellipticity (i.e. μ = 3) as a model which is closely related to the counterexample of Section 4.4. For the moment we replace Assumption 4.1 by Assumption 4.22. There are constants c1 , . . . , c7 such that for all x ∈ Ω, for all P , U , V ∈ RnN and for γ = 1, . . . , n i) the variational integrand f = f (x, P ) is of class C 2 Ω × RnN and any of the derivatives occurring below exists; ii) ∇P f (x, P ) ≤ c1 ; − 3 − 1 iii) c2 1 + |P |2 2 |U |2 ≤ DP2 f (x, P )(U, U ) ≤ c3 1 + |P |2 2 |U |2 ; iv) ∂γ ∇P f (x, P ) ≤ c4 ; v) ∂γ ∂γ ∇P f (x, P ) ≤ c5 ; vi) ∂γ DP2 f (x, P )(U, V ) ≤ c6 DP2 f (x, P )(U, V ) +

c7 |U | |V |. 1 + |P |2

Moreover, we assume that the variational integrand f = f (x, P ) is of linear growth in P , uniformly with respect to x, i.e. a |P | − b ≤ f (x, P ) ≤ A |P | + B holds with constants which are not depending on x. Remark 4.23. Maybe, assumption vi) needs some brief comment: if we want to include integrands of the type f (x, P ) = g(α(x)P ) with some scalar function α in our considerations, then we cannot expect that ∂γ DP2 f and DP2 f define equivalent bilinear forms on RnN . However, the admissible perturbation on the right-hand side of vi) in particular gives a suitable approach to our example of Section 4.4 (compare Remark 4.43). We now claim Theorem 4.24. Theorem 4.14 remains true if Assumption 4.1 is replaced by Assumption 4.22.

114


Proof. With the regularization introduced in Section 4.2.1 we first note that the arguments from duality theory remain unchanged (see, for instance, [ET]) and that we again have Remark 4.18. Moreover, Theorem A.11 is not affected by an additional x-dependence, hence we merely have to establish Theorem 4.21 with Assumption 4.22. To this purpose we recall the Euler equation in its non-differentiated form (25) ∇P fδ (x, ∇uδ ) : ∇ϕ dx = 0 for all ϕ ∈ C0∞ Ω; RN , Ω

which is of the same type as before. Now, the estimates (21)–(24) only depend on (25) if we take the following observation into account. To prove Remark 4.2, i), we now replace f (x, P ) by fx0 := f (x, P ) − ∇P f (x0 , 0) : P . Note that the constants c2 –c7 occurring in Assumption 4.22 remain unchanged. Moreover, Assumption 4.22, ii), holds uniformly with respect to x0 for any fx0 as above. As a consequence, we obtain |P | 3 (1 + ρ2 )− 2 dρ . ∇P f (x0 , P ) : P ≥ c |P | 0

Then we choose r sufficiently small such that we have Remark 4.2, i), on Br (x0 ). Summarizing these results, the theorem is proved if we can show that DP2 fδ (x, ∇uδ )(∂γ ∇uδ , ∂γ ∇uδ ) η 2 dx ≤ c(η) (26) B2r (x0 )

for any B2r (x0 ) Ω and all η ∈ C0∞ (B2r (x0 )), 0 ≤ η ≤ 1. At this point the differentiated form of (25) is needed which reads as 2 ∂γ ∇P fδ (x, ∇uδ ) : ∇ϕ dx = 0 (27) DP fδ (x, ∇uδ )(∂γ ∇uδ , ∇ϕ) dx + Ω

Ω

for all ϕ ∈ C0∞ (Ω; RN ). Once more the starting integrability of uδ is good enough to take ϕ = η 2 ∂γ uδ (η as above) as an admissible test-function in (27). As a result we obtain DP2 fδ (x, ∇uδ )(∂γ ∇uδ , ∂γ ∇uδ ) η 2 dx B2r (x0 ) = −2 DP2 fδ (x, ∇uδ )(∂γ ∇uδ , ∂γ uδ ⊗ ∇η) η dx B2r (x0 ) −2 (∂γ ∇P fδ )(x, ∇uδ ) : ∂γ uδ ⊗ ∇η η dx B2r (x0 ) − (∂γ ∇P fδ )(x, ∇uδ ) : ∂γ ∇uδ η 2 dx B2r (x0 )

=: I + II + III .

(28)


By Assumption 4.22, iv), we have |II| ≤ c(η)

115

|∇uδ | dx ≤ c .

B2r (x0 )

The first integral on the right-hand side of (28) is handled with Young’s inequality for ε > 0 sufficiently small |I| ≤ ε DP2 fδ (x, ∇uδ )(∂γ ∇uδ , ∂γ ∇uδ ) η 2 dx B2r (x0 ) −1 +c ε |DP2 fδ (x, ∇uδ )| |∇η|2 |∇uδ |2 dx . B2r (x0 )

Here the second integral on the right-hand side is uniformly bounded, the first one can be absorbed on the left-hand side of (28), hence it remains to find an upper bound for III. We perform an integration by parts to obtain III = (∂γ ∂γ ∇P fδ )(x, ∇uδ ) : ∇uδ η 2 dx B2r (x0 ) + (∂γ DP2 fδ )(x, ∇uδ )(∂γ ∇uδ , ∇uδ ) η 2 dx B2r (x0 ) + (∂γ ∇P fδ )(x, ∇uδ ) : ∇uδ ∂γ η 2 dx B2r (x0 )

=: III1 + III2 + III3 . Assumption 4.22, v), shows that |III1 | is bounded independent of δ, the uniform estimate for |III3 | again follows from iv) of Assumption 4.22. Finally, for the consideration of |III2 | we make use of Assumption 4.22, vi), which together with Young’s inequality gives for ε > 0 (note that the γ-derivative of the δ-part vanishes) |III2 | ≤ c B2r (x0 )

|DP2 f (x, ∇uδ )(∂γ ∇uδ , ∇uδ )| η 2 dx

+c

1 + |∇uδ |2

−1

|∇2 uδ | |∇uδ | η 2 dx

B2r (x0 ) ≤ cε B2r (x0 ) −1

DP2 f (x, ∇uδ )(∂γ ∇uδ , ∂γ ∇uδ ) η 2 dx (29)

+c ε

B2r (x0 )

+c ε

1 + |∇uδ |2

B2r (x0 ) −1

DP2 f (x, ∇uδ )(∇uδ , ∇uδ ) η 2

+c ε

B2r (x0 )

− 32

1 + |∇uδ |2

dx

|∇2 uδ |2 η 2 dx

− 12

|∇uδ |2 η 2 dx .

116


Here, on account of the ellipticity Assumption 4.22, iii), the third integral on the right-hand side is estimated by the first one which in return is absorbed on the left-hand side of (28). The remaining two integrals on the right-hand side of (29) are handled with the linear growth of f which, by assumption, is uniformly with respect to x (we also recall the second inequality of Assumption 4.22, iii)). Thus, the theorem is proved. As mentioned above, we will come back to the x-dependent situation in Section 4.4. 4.2.3 Lp -estimates in the case μ < 3 From now on we assume a slightly stronger ellipticity condition, i.e. we consider the case μ < 3. Moreover, we again have the additional structure in the vectorial setting N > 1. Then it is possible to modify the arguments of Section 4.2.2.1 such that the results obtained there may be iterated. This gives uniform Lp -estimates in the sense of Theorem 4.25. Suppose that μ < 3, that we have Assumption 4.1, Assumption 4.11 and that (2), Section 4.1, is satisfied. Then for any 1 < p < ∞ and for any Ω Ω there is a constant c(p, Ω ), which does not depend on δ, such that ∇uδ Lp (Ω ;RnN ) ≤ c(p, Ω ) < ∞ . Remark 4.26. As an immediate consequence we can find a generalized min1 imizer u∗ ∈ M which is of class Wp,loc (Ω; RN ) for any 1 < p < ∞. Proof of Theorem 4.25. Fix a ball Br0 (x0 ) Ω and assume that there is a real number α0 ≥ 0 such that (uniformly with respect to δ) 1+α0 α 1+ 0 2 Γδ dx + δ Γδ 2 dx ≤ c . (30) Br0 (x0 )

Br0 (x0 )

Note that by (13) this assumption is true for α0 = 0. Next define α = α0 +3−μ α and choose ϕ = uδ Γδ2 η 2 , η ∈ C0∞ (Br0 (x0 )), 0 ≤ η ≤ 1, η ≡ 1 on Br0 /2 (x0 ), 2 1 ∩ W∞,loc (Ω; RN ), hence |∇η| ≤ c/r0 . As outlined above, uδ is of class W2,loc ϕ is admissible in (14) with the result α α 2 2 ∇f (∇uδ ) : ∇uδ Γδ η dx + δ |∇uδ |2 Γδ2 η 2 dx Br0 (x0 ) Br0 (x0 ) α−1 α 2 2 2 ≤ c(α) Γδ |∇ uδ | η dx + c(α) δ Γδ2 |∇2 uδ | η 2 dx (31) B (x ) Br0 (x0 ) r0 0 α+1 α +c Γδ2 |∇η 2 | dx + c δ Γδ 2 |∇η 2 | dx . Br0 (x0 )

Br0 (x0 )

Here Assumption 4.11 and the boundedness of ∇f (compare Assumption 4.1, ii)) are used. Analogous to Section 4.2.2.1, the left-hand side of (31) is estimated with the help of Remark 4.2, i):


1+α 2

l.h.s. ≥ ν1 B

r0

(x0 )

Br0 (x0 )

1+ α 2

+δ Br0 (x0 )

α

η dx − ν2 2

Γδ

Γδ

117

Γδ2 η 2 dx α

η 2 dx − δ Br0 (x0 )

Γδ2 η 2 dx .

The right-hand side of (31) is handled via (fix ε > 0 and use Young’s inequality) r.h.s ≤ c Br0 (x0 )

1+α − 1+α η 2 ε Γδ 2 + ε−1 Γδ 2 Γδα−1 |∇2 uδ |2 dx

+c Br0 (x0 )

+c δ Br0 (x0 )

1+α

− 1+α 2

ε Γδ 2 η 2 + ε−1 Γδ

2 dx Γα |∇η| δ

1+ α −1− α 2 2 dx η 2 ε Γδ 2 + ε−1 Γδ 2 Γα |∇ u | δ δ

+c δ Br0 (x0 )

1+ α 2

ε Γδ

−1− α 2

η 2 + ε−1 Γδ

2 dx . Γ1+α |∇η| δ

Hence, absorbing terms, (31) yields

1+α 2

Br0 /2 (x0 )

Γδ

1+ α 2

dx + δ Br0 /2 (x0 )

Γδ

dx

- α−3 ≤ c η 2 Γδ 2 |∇2 uδ |2 dx B (x ) . r0 0 α−1 α 2 2 + Γδ 2 |∇η| dx + Γδ2 η dx Br0 (x0 ) Br0 (x0 ) - α −1 + cδ η 2 Γδ2 |∇2 uδ |2 dx Br0 (x0 ) . α α 2 2 2 2 + Γδ |∇η| dx + Γδ η dx

=: c

Br0 (x0 ) 3 +

6 +

i=1

i=4

Ii + c δ

(32)

Br0 (x0 )

Ii .

Starting with I1 , we recall that by definition μ + α − 3 = α0 ≥ 0, thus Assumption 4.1, iii), and Lemma 4.19, i), give

−μ 2

I1 = Br0 (x0 )

η 2 Γδ

≤c Br0 (x0 )

μ+α−3 2

|∇2 uδ |2 Γδ

dx

α0 D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ Γδ2 η 2 dx

≤ c(r0 ) Br0 (x0 )

− 12

Γδ

α 1+ 0 + δ Γδ 2 dx ≤ c ,

118


where the last inequality is due to the assumption (30). An upper bound (not depending on δ) for I3 is found since we may assume without loss of generality that μ ≥ 2. This clearly proves I2 to be bounded independent of δ as well. Studying I4 let us first assume that α ≤ 2. Then, again by Lemma 4.19, i), D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ η 2 dx δ I4 ≤ Br0 (x0 ) 1 −2 ≤ c(r0 ) Γδ + δ Γδ dx ≤ c . Br0 (x0 )

In the case α > 2, Lemma 4.19, i), gives α −1 δ I4 ≤ D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ Γδ2 η 2 dx Br0 (x0 ) 1 − 1+ α −1 Γδ 2 + δ Γ δ 2 ≤ c(r0 ) dx ≤ c , Br0 (x0 )

where we once more recall (30) and observe that α2 −1 = (α0 +1−μ)/2 ≤ α0 /2. This condition trivially bounds δI5 and δI6 (independent of δ) and we have proved with (32): suppose that (30) holds for some given r0 > 0 and α0 ≥ 0. Then there is a constant, independent of δ, such that 1+α0 +3−μ α +3−μ 1+ 0 2 2 Γδ dx + δ Γδ dx ≤ c . (33) Br0 /2 (x0 )

Br0 /2 (x0 )

We now claim that for any m ∈ N there is a constant c(m), independent of δ, such that 1+m(3−μ) m(3−μ) 1+ 2 2 Γδ dx + δ Γδ dx ≤ c . (34) Br0 /2m (x0 )

Br0 /2m (x0 )

In fact, as mentioned above, α0 = 0 is an admissible choice to obtain (34) from (33) in the case m = 1. Next assume by induction that (34) is true for some m ∈ N. Then we may take α0 = m(3 − μ) in (30) and (33) gives (34) with m replaced by m + 1, thus the claim is proved. Obviously, this implies Theorem 4.25. Remark 4.27. If we omit the structure condition in the vectorial setting, then analogous arguments prove higher integrability up to a finite number 1 < p(μ). 4.2.4 A priori gradient bounds Here the DeGiorgi-type arguments as outlined in Section 3.3.3 are modified: on one hand, given Theorem 4.25, we benefit from H¨ older’s inequality. This decreases on the other hand the exponent β of iteration (see the definition of β given below). Nevertheless, it turns out that Lemma 3.26 still is applicable to obtain


119

Theorem 4.28. Consider a ball BR0 (x0 ) Ω. With the assumptions of Theorem 4.25 there is a local constant c > 0 such that for any δ ∈ (0, 1) ∇uδ L∞ (BR0 /2 ,RnN ) ≤ c . Before proving Theorem 4.28 we recall the definitions Γδ = 1 + |∇uδ |2 , A(k, r) = x ∈ Br (x0 ) : Γδ > k , Br (x0 ) Ω , k > 0 , and establish the following result. Lemma 4.29. Fix some x0 ∈ Ω and suppose that we are given radii 0 < r < R < R0 , BR0 (x0 ) Ω. Then there is a real number c, independent of r, R, R0 , k and δ, such that n Γδ − k) n−1 dx A(k,r) (35) n - n . 12 n−1 . 12 n−1 - μ c 2 2 ≤ Γδ − k) dx Γδ dx . n (R − r) n−1 A(k,R) A(k,R) Proof of Lemma 4.29. Recalling the notion w+ = max[w, 0], Sobolev’s inequality yields for all η ∈ C0∞ (BR (x0 )) such that 0 ≤ η ≤ 1, η ≡ 1 on Br (x0 ), |∇η| ≤ c/(R − r),

Γδ − k

n n−1

n + n−1 dx η Γδ − k

dx ≤ BR (x0 )

A(k,r)

n . n−1 + ∇ η Γδ − k dx

- ≤c

BR (x0 )

- ≤c

n . n−1 ∇ η Γδ − k dx

A(k,R) n n−1

≤ c I1

n n−1

+ I2

(36)

.

Here we have n n−1

I1

- := A(k,R)

- ≤

n . n−1 |∇η| Γδ − k dx

n - . 12 n−1 2 |∇η|2 Γδ − k dx

A(k,R)

c ≤ n (R − r) n−1 n

- A(k,R)

n . 12 n−1 1 dx

A(k,R)

n - . 12 n−1 2 Γδ − k dx

n . 12 n−1 1 dx ,

A(k,R)

thus I1n−1 is bounded from above by the right-hand side of (35). Estimating I2 , Lemma 4.19, ii), is needed with the result


120

n . n−1 η |∇Γδ | dx

-

n n−1

I2

:= A(k,R)

- ≤

η |∇Γδ | 2

A(k,R)

-

2

n - . 12 n−1 dx

−μ Γδ 2

n . 12 n−1 Γδ dx μ 2

A(k,R)

n . 12 n−1 2 D fδ (∇uδ ) ej ⊗ ∇η, ej ⊗ ∇η Γδ − k dx

≤ c

2

-

A(k,R)

n . 12 n−1 Γδ dx μ 2

· A(k,R)

c ≤ n (R − r) n−1

-

Γδ − k

2

n - . 12 n−1 dx

A(k,R)

n . 12 n−1 Γδ dx , μ 2

A(k,R)

hence (36) proves the lemma. We now come to the Proof of Theorem 4.28. Consider the left-hand side of (35): for any real number s > 1, H¨ older’s inequality implies

Γδ − k

2

dx =

A(k,r)

Γδ − k

n 1 n−1 s

A(k,r)

-

≤

Γδ − k

n n−1

Γδ − k

n 1 2− n−1 s

. 1s dx

A(k,r)

-

·

Γδ − k

dx

n 1 s (2− n−1 s ) s−1

. s−1 s .

A(k,r)

Thus, on account of Theorem 4.25, there is a real number c1 (s, n, BR0 (x0 )), independent of δ, -

1 s−1

c1 (s, n, BR0 (x0 )) := sup δ>0

BR0 (x0 )

Γδ

n (2s− n−1 )

. s−1 s dx < ∞,

such that

Γδ − k

2

-

dx ≤ c1 (s, n, BR0 (x0 ))

A(k,r)

Γδ − k

n n−1

. 1s dx . (37)

A(k,r)

Studying the right-hand side of (35), we fix a second real number t > 1 and applying H¨ older’s inequality once more it is established that A(k,R)

Let

1 Γδ dx ≤ A(k, R) t μ 2

- A(k,R)

μ t 2 t−1

Γδ

. t−1 t dx .


-

μ t 2 t−1

c2 (t, μ, BR0 (x0 )) := sup δ>0

Then we have

A(k,R)

BR0 (x0 )

Γδ

121

. t−1 t dx < ∞.

μ 1 Γδ2 dx ≤ c2 (t, μ, BR0 (x0 )) A(k, R) t .

(38)

Summarizing these results we arrive at

Γδ − k

2

-

(37)

dx ≤ c

A(k,r)

Γδ − k

A(k,r) (35)

≤

-

c

. 1s dx

Γδ − k n 1 A(k,R) (R − r) n−1 s n 1 . 12 n−1 - s μ 2 · Γδ dx A(k,R)

(38)

≤

n n−1

-

c n

1

(R − r) n−1 s

Γδ − k

2

n 1 . 12 n−1 s dx

(39) 2

n 1 . 12 n−1 s dx

A(k,R)

1 n 1 1 · A(k, R) 2 n−1 s t . As the next step we define for k and r < R as above the following quantities:

τ (k, r) :=

Γδ − k

2

dx ,

A(k,r)

a(k, r) := A(k, r) . With this notation, (39) can be written in the form τ (k, r) ≤

c n

1

(R − r) n−1 s

1 n 1 1 n 1 1 τ (k, R) 2 n−1 s a(k, R) 2 n−1 s t .

(40)

Given two real numbers h > k > 0, we now observe 2 Γδ − k (h − k)−2 dx , dx ≤ a(h, R) = A(h,R)

A(h,R)

thus we get for h > k > 0 a(h, R) ≤

1 τ (k, R) . (h − k)2

With (40) and (41) it is proved that for h > k > 0 we have the estimate

(41)


122

τ (h, r) ≤ ≤

c (R − r)

n 1 n−1 s

1

c (R − r)

1 n 1 τ (h, R) 2 n−1 s

n 1 n−1 s

(h − k)

n 1 1 n−1 s t

1 (h − k)

n 1 1 n−1 s t

1 n 1 1 τ (k, R) 2 n−1 s t

1 n 1 (1+ 1t ) τ (k, R) 2 n−1 s .

Now s and t are chosen sufficiently close to 1 (depending on n) such that . 1 1 n 1 1+ =: β > 1 . 2 n−1 s t With this choice of s and t we additionally let α :=

n 11 > 0, n−1 s t

γ :=

n 1 > 0. n−1 s

Thus we are in the situation of Lemma 3.26. Applying this lemma we have 2 τ (d, R0 /2) = Γδ − d dx = 0 , A(d,R0 /2)

and, as a consequence, Γδ ≤ d

on BR0 /2 (x0 ) .

(42)

Here the quantity d is uniformly bounded with respect to δ if and only if there is a constant (independent of δ) such that Γ2δ dx ≤ c . τ (0, R0 ) = BR0 (x0 )

This fact however was proved in Theorem 4.25 and the a priori estimate Theorem 4.28 follows from (42). Remark 4.30. With Theorem 4.28 we also have a complete Proof of Theorem 4.16. This was already discussed in Section 4.1.3. Note that Proposition 4.10 again can be replaced by the arguments given in Section 2.3.2.

4.3 Two-dimensional problems To finish this chapter we study two-dimensional μ-elliptic variational problems with linear growth. It turns out that (assuming the usual structure condition if N > 1) in this case no boundedness condition is necessary to find the unique (up to a constant) solution of the variational problem f (∇w) dx + f∞ (u0 − w) ⊗ ν dHn−1 → min in W11 Ω; RN (P ) Ω

∂Ω

4.3 Two-dimensional problems

123

in the limit case μ = 3. We then reduce the two-dimensional (unbounded) case to the setting of Section 4.2 and get, if n = 2, the validity of Theorem 4.16 (μ < 3) in the most general setting. To be precise: from now on we consider boundary data u0 ∈ W21 (Ω; RN ), where again the case u0 ∈ W11 (Ω; RN ) is handled as outlined in Remark 2.5 (compare [Bi5]). We are going to prove Theorem 4.31. Suppose that Assumption 4.1 holds and that we have in addition (2), Section 4.1, in the vectorial case N > 1. Moreover, consider the limit case μ = 3. Then, if n = 2, there is a generalized minimizer u∗ ∈ M such that i) We have ∇s u∗ ≡ 0, hence ∇u∗ ≡ ∇a u∗ . ii) For any Ω Ω there is a constant c(Ω ) satisfying |∇u∗ | ln 1 + |∇u∗ |2 dx ≤ c(Ω ) < ∞ . Ω

iii) The particular minimizer u∗ is of class W11 (Ω; RN ) and (up to a constant) the unique solution of the variational problem (P ). Theorem 4.32. Suppose that Assumption 4.1 is valid with μ < 3. In the case N > 1 we additionally assume (2) and (3) as stated in Section 4.1. Then, if n = 2, we have i) Each generalized minimizer u ∈ M is in the space C 1,α (Ω; RN ) for any 0 < α < 1. ii) For u, v ∈ M we have ∇u = ∇v, i.e. generalized minimizers are unique up to a constant. 4.3.1 Higher local integrability in the limit case Here we are going to establish uniform local higher integrability of the sequence {uδ } (see Section 4.2.1) in the limit case μ = 3. Let us, for the moment, concentrate on the scalar case N = 1. Then we have the following assertion Lemma 4.33. Suppose that the assumptions of Theorem 4.31 hold in the twodimensional scalar case n = 2, N = 1, and let {uδ } denote the regularization introduced above. Moreover, fix a ball Br (x0 ) satisfying B2r (x0 ) Ω. Then there is a positive number c = c(r), independent of δ, such that for all η ∈ C0∞ (B2r (x0 )), 0 ≤ η ≤ 1, 1 1 + |∇uδ |2 2 |uδ − (uδ )2r |2 η 2 dx B2r (x0 ) +δ |∇uδ |2 |uδ − (uδ )2r |2 η 2 dx ≤ c . B2r (x0 )

Here (uδ )2r denotes the mean value of uδ on B2r (x0 ).

124


Remark 4.34. i) Following the proof of Theorem 4.36 below, it becomes obvious that this estimate is exactly the one which is needed to reach the limit case μ = 3. ii) Inequality (43) given below is the main reason why the results in two dimensions are better than the ones stated above for n ≥ 3. Proof of Lemma 4.33. Note that in the two-dimensional case n = 2 we have by Sobolev-Poincarè’s inequality 12

|uδ − (uδ )2r | dx 2

≤ c1

B2r (x0 )

|∇uδ | dx ≤ c2

(43)

B2r (x0 )

for some positive constants c1 , c2 which are not depending on δ (recall Remark 4.2, ii), and (13), Section 4.2.1). Since uδ is a solution of the approximative problem we have uδ ∈ Wt2 (B2r (x0 )) for any t < ∞. This was outlined in Section 3.3.1. As a result, ϕ = (uδ − (uδ )2r )3 η 2 , η ∈ C0∞ (B2r (x0 )), 0 ≤ η ≤ 1, is admissible in the Euler equation (see (14), Section 4.2.1) and we obtain 3

∇f (∇uδ ) ∇uδ |uδ − (uδ )2r |2 η 2 dx B2r (x0 ) +3 δ |∇uδ |2 |uδ − (uδ )2r |2 η 2 dx B2r (x0 ) 3 = −2 ∇fδ (∇uδ ) ∇η η uδ − (uδ )2r dx . B2r (x0 )

Remark 4.2, i), (43) and the boundedness of |∇f | imply

B2r (x0 )

1 1 + |∇uδ |2 2 |uδ − (uδ )2r |2 η 2 dx +δ |∇uδ |2 |uδ − (uδ )2r |2 η 2 dx

(44)

B2r (x0 )

≤ c (1 + I1 + I2 ) , where the constant c again is not depending on δ, and I1 , I2 are given by |uδ − (uδ )2r |3 η |∇η| dx ,

I1 = B2r (x0 )

|∇uδ | |uδ − (uδ )2r |3 η |∇η| dx .

I2 = δ B2r (x0 )

Estimating I1 we observe (using (43), H¨ older’s inequality, Sobolev-Poincarè’s inequality and Young’s inequality for some sufficiently small number ε > 0)


125

12

I1 ≤

|uδ − (uδ )2r |4 η 2 dx B2r (x0 )

12

·

|uδ − (uδ )2r |2 |∇η|2 dx B2r (x0 )

∇ |uδ − (uδ )2r |2 η dx

≤c B2r (x0 )

≤c

|uδ − (uδ )2r | |∇uδ | η dx

1+

(45)

B2r (x0 )

1 ε |uδ − (uδ )2r |2 1 + |∇uδ |2 2 η 2

≤ c 1+ B2r (x0 )

+ε

−1

1 1 + |∇uδ |2 2 dx .

Here again c denotes some positive local constant which is not depending on δ. Note that the “ε”-part on the right-hand side of (45) can be absorbed (for ε sufficiently small) on the left-hand side of (44), whereas the remaining integral is uniformly bounded with respect to δ. To find an estimate for I2 we recall that |∇uδ |2 dx → 0 as δ → 0 . δ Ω

This was proved in (9), Section 2.1.2. As a consequence, we have with suitable local constants and for ε > 0 sufficiently small 12 |∇uδ |2 dx

I2 ≤ c δ 1

B2r (x0 )

12 |uδ − (uδ )2r |6 η 2 dx

B2r (x0 )

∇ |uδ − (uδ )2r |3 η dx

≤ cδ 2 B2r (x0 )

≤ cδ

1 2

|uδ − (uδ )2r |2 |∇uδ | η dx + B2r (x0 )

1

1

εδ 2 |uδ − (uδ )2r |2 |∇uδ |2 η 2

B2r (x0 )

+ε−1 δ

− 12

|uδ − (uδ )2r |3 dx B2r (x0 )

≤ cδ 2

|uδ − (uδ )2r |2 dx +

|uδ − (uδ )2r |3 dx

B2r (x0 )

=: c

3 + i=1

I2i .

(46)


126

Note that this estimate is not sharp in the sense that the constant occurring in the second line in fact tends to zero as δ → 0. Anyhow, I21 can be absorbed on the left-hand side of (44), whereas I22 is uniformly bounded with respect to δ. I23 is estimated with the help of (43), H¨ older’s and Sobolev-Poincarè’s inequality I23

=δ

1 2

|uδ − (uδ )2r |3 dx B

≤δ

1 2

2r

(x0 )

12 |uδ − (uδ )2r |4 dx

B2r (x0 )

·

12

|uδ − (uδ )2r |2 dx B2r (x0 ) 1

∇|uδ − (uδ )2r |2 dx

≤ cδ 2 B

1

(47)

2r

(x0 )

≤ cδ 2 B2r (x0 )

12 |∇uδ |2 dx

12 |uδ − (uδ )2r |2 dx

B2r (x0 )

≤c. If we recall that the “ε–terms” are absorbed on the left-hand side of (44) (as described above), then (45)–(47) complete the proof of Lemma 4.33. Remark 4.35. Going through the proof of Lemma 4.33 we see that the assertion is not depending on the exponent μ of ellipticity. The above lemma now yields uniform higher local integrability of |∇uδ | in the scalar case. Theorem 4.36. Consider the two-dimensional scalar case n = 2, N = 1 together with the general assumption of Theorem 4.31. If B2r (x0 ) Ω then there exists a local constant c, independent of δ, such that 1 1 + |∇uδ |2 2 ln 1 + |∇uδ |2 dx ≤ c . Br (x0 )

Proof. We let ωδ = ln(1 + |∇uδ |2 ) and choose ϕ = (uδ − (uδ )2r ) ωδ η 2 , η ∈ C0∞ (B2r (x0 )), 0 ≤ η ≤ 1, η ≡ 1 on Br (x0 ). Again ϕ is seen to be admissible in the Euler equation (14), Section 4.2.1, and we obtain


127

∇f (∇uδ ) ∇uδ ωδ η dx + δ

|∇uδ |2 ωδ η 2 dx

2

B2r (x0 )

B2r (x0 )

∇f (∇uδ ) ∇ωδ uδ − (uδ )2r η 2 dx

= − B2r (x0 )

∇f (∇uδ ) ∇η η uδ − (uδ )2r ωδ dx

−2 B2r (x0 )

∇uδ ∇ωδ uδ − (uδ )2r η 2 dx

−δ B2r (x0 )

∇uδ ∇η η uδ − (uδ )2r ωδ dx

−2 δ B2r (x0 )

=:

4 +

Ii .

i=1

Similar to the proof of Lemma 4.33, a lower bound for the first integral on the left-hand side is given by Remark 4.2, i), thus

B2r (x0 )

1 + |∇uδ |

2

12

|∇uδ |2 ωδ η 2 dx B2r (x0 ) 4 + ωδ η 2 dx + |Ii | . 2

ωδ η dx + δ

≤ c B2r (x0 )

(48)

i=1

Clearly B2r (x0 ) ωδ η 2 dx is uniformly bounded with respect to δ and in order to find an estimate for I1 we observe |∇ωδ |2 ≤

4 |∇2 uδ |2 . 2 1 + |∇uδ |

This, together with Lemma 4.33, implies (again we make use of the fact that |∇f | is bounded) |I1 | ≤ c

|uδ − (uδ )2r | |∇ωδ | η 2 dx B

2r

(x0 )

≤c

B2r (x0 )

·

12 1 − 1 + |∇uδ |2 2 |∇ωδ |2 η 2 dx

B2r (x0 )

≤c

B2r (x0 )

12 1 1 + |∇uδ |2 2 |uδ − (uδ )2r |2 η 2 dx

12 3 − 1 + |∇uδ |2 2 |∇2 uδ |2 η 2 dx .

128


Here the right-hand side is bounded through the Caccioppoli-type inequality stated in Lemma 3.19 or Lemma 4.19, i). Note that we exactly reach the limit case μ = 3. Next, |uδ − (uδ )2r |2 + η 2 |∇η|2 ωδ2 dx ≤ c |I2 | ≤ c B2r (x0 )

is immediately verified and

|∇uδ |2 |uδ − (uδ )2r |2 η 2 + |∇ωδ |2 η 2 dx B2r (x0 ) −1 1+δ 1 + |∇uδ |2 |∇2 uδ |2 η 2 dx

|I3 | ≤ c δ ≤c

B2r (x0 )

≤c again follows from Lemma 4.33 and Lemma 4.19, i). Thus, together with |I4 | ≤ c δ |∇uδ |2 |uδ − (uδ )2r |2 η 2 + |∇η|2 ωδ2 dx ≤ c , B2r (x0 )

the theorem is proved recalling (48) and since the constants occurring above are not depending on δ. Let us now turn our attention to the vectorial setting N > 1. Theorem 4.37. Consider the case N > 1 and suppose that the assumptions of Theorem 4.31, in particular (2), Section 4.1, hold. Then we still have the claims of Theorem 4.36. Remark 4.38. As outlined in the previous section, this result is combined with Theorem A.11 to obtain a Proof of Theorem 4.31. Proof of Theorem 4.37. The theorem is established once the following claims are verified (we keep the notation introduced above). i) ϕ = |uδ − (uδ )2r |2 (uδ − (uδ )2r ) η 2 is admissible in the Euler equation (14), Section 4.2.1 (this test-function is used to prove Lemma 4.33). ii) This choice of ϕ implies (44). iii) The test-function ϕ = uδ − (uδ )2r ωδ η 2 also is admissible (this is necessary for the proof of Theorem 4.36). If i)–iii) are valid, then the remaining arguments given in the proofs of Lemma 4.33 and Theorem 4.36 can be carried over to the vectorial setting without any changes. ad i) & iii). We already have noted several times (see, for instance, (28), 2 1 Section 3.3.3) that uδ is of class W2,loc (Ω; RN ) ∩ W∞,loc (Ω; RN ). This immediately gives i) and iii).


129

ad ii). Here we cannot refer to Remark 4.2 without a further comment since the structure condition is violated if we pass to the energy density f¯ = f (Z) − ∇f (0) : Z. However, the representation f (Z) = g(|Z|2 ) also implies ∇f (0) = 0 , and then we may follow the arguments of Remark 2 4.2. Thus, we obtain ∇f (Z) : Z ≥ 0 and (with the notation fδ (Z) = gδ |Z| ) gδ |Z|2 ≥ 0

for all Z ∈ R2N .

We next let ψ = |uδ − (uδ )2r |2 (uδ − (uδ )2r ) and have almost everywhere ∇fδ (∇uδ ) : ∇ψ = 2 gδ |∇uδ |2 ∇uδ : ∇ψ # = 2 gδ |∇uδ |2 ∂α uiδ ∂α uiδ |uδ − (uδ )2r |2 i j j j i i + ∂α uδ uδ − (uδ )2r 2 ∂α uδ uδ − (uδ )2r

$

≥ 2 gδ |∇uδ |2 ∂α uiδ ∂α uiδ |uδ − (uδ )2r |2 = ∇fδ (∇uδ ) : ∇uδ |uδ − (uδ )2r |2 . This implies (44) exactly in the same way as above.

4.3.2 The case μ < 3 It remains to give a Proof of Theorem 4.32. We proceed in three steps: first we fix a L1 -cluster point u∗ ∈ M of the regularizing sequence {uδ } and use Theorem 4.31 to define a suitable local auxiliary variational problem. Here we find uniform local gradient estimates according to Theorem 4.28. Then, the auxiliary solutions are modified and extended to the whole domain Ω. We obtain a sequence {wm } where it turns out that L1 -cluster points w∗ are generalized minimizers of the original problem, hence elements of the set M. Finally, the duality relation holds almost everywhere both for u∗ and for w∗ which gives the theorem. Step 1. From now on we suppose that the assumptions of Theorem 4.32 are valid. We fix a L1 -cluster point u∗ of the regularizing sequence {uδ } and recall that u∗ is of class W11 (Ω; RN ) (compare Section 4.3.1). Moreover, we fix x0 ∈ Ω and write (with a slight abuse of notation) u∗ (r, θ) = u∗ (x0 + reiθ ). We assume that B2R0 (x0 ) Ω and observe that

130


R0

2π

0

0

∗ R0 2π ∂u |∇u∗ | dθ r dr ≤ c < ∞ . ∂θ dθ dr ≤ 0 0

Hence, there exists a radius R0 /2 ≤ R ≤ R0 such that 2π ∗ ∂u (R, θ) dθ ≤ c < ∞ . ∂θ 0

(49)

Next we pass to a smooth sequence {um }, um ∈ C ∞ (Ω; RN ), with the property (50) um → u∗ in W11 Ω; RN as m → ∞ , hence it is possible to estimate

R0

R0

R0

2π

hm (r) dr := 0

0

≤

0

0 2π

∂(um − u∗ ) dθ dr ∂θ |∇(um − u∗ )| dθ r dr

m→∞

→

0.

0

Thus, hm (r) → 0 in L1 ((0, R0 )) as m → ∞ and we may assume in addition to (49) that R is chosen to satisfy 2π ∂um (R, θ) dθ ≤ c < ∞ , (51) ∂θ 0 where the constant c does not depend on m. As a consequence of (51) it is finally established: there is a radius R ∈ (R0 /2, R0 ) and real number K > 0 such that for all m ∈ N um |∂B (x ) ≤ K . (52) 0 R We have found suitable boundary data to consider the variational problem

δ f (∇w) dx + |∇w|2 dx 2 BR (x0 ) BR (x0 ) ◦ → min in um + W21 BR (x0 ); RN .

Jδ w, BR (x0 ) :=

(Pδm )

If δ = δ(m) is chosen sufficiently small and if we denote by vm the unique solution of (Pδm ), then (53) Jδ(m) vm , BR (x0 ) ≤ Jδ(m) um , BR (x0 ) ≤ c holds with a constant c not depending of m. Moreover, by (52), we find (citing for example the maximum principle given in [DLM] or the convex hull property discussed in the next chapter) vm L∞ (BR (x0 );RN ) ≤ K .

(54)


131

At this point we recall that the a priori gradient estimates established in Theorem 4.28 only depend on the data and the constants occurring on the right-hand side of (53) and (54), respectively. As a result, a real number c > 0, independent of m, is found such that ∇vm L∞ (BR/2 (x0 );R2N ) ≤ c .

(55)

Step 2. Given u∗ , um and vm as above, we choose η ∈ C ∞ (BR (x0 )), η ≡ 1 on 1 : BR (x0 ) → RN , BR (x0 ) − B3R/4 (x0 ), η ≡ 0 on BR/2 (x0 ), and let wm 1 wm := vm + η (u∗ − um ) ,

1 ∗ wm |∂BR (x0 ) = u|∂BR (x0 ) .

hence

1 We then claim that wm provides a J|BR (x0 ) -minimizing sequence with respect ∗ to the boundary data u|BR (x0 ) : in fact, (50) implies as m → ∞

∗ f (∇um ) − f (∇u ) dx ≤ c |∇um − ∇u∗ | dx → 0 , BR (x0 ) BR (x0 ) and if we decrease δ (if necessary), then we obtain from the minimality of vm f (∇vm ) dx BR (x0 )

Jδ(m) vm , BR (x0 ) ≤ Jδ(m) um , BR (x0 ) m→∞ → f (∇u∗ ) dx . ≤

(56)

BR (x0 )

Moreover, we have 1 f (∇wm ) − f (∇vm ) dx BR (x0 )

≤

c

∇ η (u∗ − um ) dx

BR (x0 ) m→∞

→ 0,

which, together with (56) and the minimality of u∗ (recall that u∗ ∈ W11 (Ω; RN ) is a local J-minimizer) implies m→∞ 1 f (∇wm ) dx → f (∇u∗ ) dx , (57) BR (x0 )

BR (x0 )

i.e. our claim is proved. Now once more Lemma B.5 on local comparison functions comes into play: the trace of each wm coincides with the trace of u∗ on ∂BR (x0 ). Further, we 1 } to a J-minimizing have (57). Thus, it is possible to extend the sequence {wm ◦

sequence {wm } from u0 + W11 (Ω; RN ). Hence, if w∗ denotes a weak cluster

132


point of {wm }, then w∗ ∈ M. Step 3. Finally we recall that u∗ is a weak cluster point of the regularizing sequence {uδ }. Hence, the results of Sections 2.3.1 and 2.3.2 apply and we find an open set Ω0 of full measure such that u∗ ∈ C 1,α Ω0 ; RN . Furthermore, if σ denotes the unique solution of the dual variational problem, then σ = ∇f (∇u∗ ) in Ω0 . At this point we like to emphasize that the variation of σ leading to Theorem A.9 is of local character. In particular, the assertion remains valid for any generalized minimizer v ∗ ∈ M with Ω replaced by the open set Ω0 , where σ is known to be a continuous function taking values in Im(∇f ). Hence we also have σ = ∇f (∇v ∗ ) in Ω0 . Since w∗ ∈ M we obtain ∇w∗ = ∇u∗

almost everywhere .

On the other hand, recall that 1 wm |BR/2 (x0 ) = wm |BR/2 (x0 ) = vm |BR/2 (x0 ) ,

hence the a priori estimate (55) yields ∇u∗ L∞ (BR/2 (x0 );R2N ) ≤ c . Note u∗ really is Lipschitz continuous since u∗ ∈ W11 (Ω; RN ), in particular ∇s u∗ ≡ 0, is due to the previous considerations. Keeping Remark 4.30 in mind we have proved Theorem 4.32.

4.4 A counterexample The studies of linear growth variational problems are completed with an example which shows the sharpness of our results. The idea originates from [GMS1], Example 3.2, where the authors restrict themselves to the one-dimensional situation. We follow the proposal of Giaquinta, Modica and Souˇcek and give a rigorous proof that the arguments extend to higher-dimensional annuli Ω. What is more, the example given in [GMS1] is degenerated which is not the case in the modification outlined below. As a consequence, we precisely can verify the assumptions of Section 4.2.2.2 (note that our arguments rely on a smooth x-dependence) with the exception that we now have μ > 3.

4.4 A counterexample

133

The general setting is the following: let N = 1, n = 2 and |x| = x21 + x22 = r. We fix some positive numbers 0 < ρ1 < ρ2 , ρ := (ρ1 + ρ2 )/2 and choose Ω := x ∈ R2 : ρ1 < r < ρ2 . Moreover, α: Ω → R is defined by α(r) := 1 + γ |r − ρ|2 , where the positive parameter γ is chosen later on (see (62) and (73)). If k > 2 is fixed, then the energy density under consideration is given by ⎧ 1 ⎪ ⎪ ⎨ 1 + α(r) |P |k k if |P | > ε , f (x, P ) = f (r, P ) = ⎪ ⎪ ⎩ h(r, P ) if |P | ≤ ε . Here h(r, P ) is chosen such that f (x, P ) is strictly convex, non-degenerate in P and such that f (x, P ) of class C 2 (Ω × R2 ). For an explicit construction we may consider ⎧ 1 ⎪ ⎪ ⎨ 1 + |P |k k if |P | > ε , f˜(P ) = ⎪ ⎪ ˜ ) ⎩ h(P if |P | ≤ ε , together with the Ansatz 1 ˜ ) = a + b 1 + |P |l l + c |P |2 , h(P where a, b and c are suitable constants and l > k. The requirement that f˜ is of class C 2 (R2 ) in particular implies

b = 1 + |ε|

k

k1 −2

1 l 2− l

1 + |ε|

|ε|

k−l

k − 2 − |ε|k > 0, l − 2 − |ε|l

# $ k 1 1 k−2 k − 2 − |ε| −2 1 + |ε|k − 1 + |ε|l > 0. c = |ε| 1 + |ε|k k 2 k − 2 − |ε|l 1/k ˜ We then let h(r, P ) := h([α(r)] P ). Finally, the choice of the second parameter 0 < ε < 1 will be made in the inequality (70) below.

Theorem 4.39. With the above notation, the variational problem ◦ J[w] := f (x, ∇w) dx → min in u0 + W11 (Ω) Ω

does not admit a generalized minimizer v ∈ M of class W11 (Ω) if u0 is supposed to satisfy u0 (ρ1 ) = −a and u0 (ρ2 ) = a for a constant a > 0 sufficiently large (see (74)). Here and in the following – again with a slight abuse of notation – we write u(x1 , x2 ) = u(r) whenever u is merely depending on |x|.

134


Remark 4.40. i) Note that the ellipticity exponent of f is given by μ = k + 1 > 3, hence we really obtain an example on the sharpness of our results. ii) Moreover, it should be emphasized that the boundary values u0 may be chosen as a function of class C ∞ (Ω). Proof of Theorem 4.39. Assume by contradiction that v ∈ W11 (Ω) is a generalized minimizer. Then the proof of our assertion splits into three steps. Step 1. First of all we note that, by the symmetry of the problem and with the obvious meaning of notation (after introducing polar coordinates), we have v(r, ϕ) = v(r) .

(58)

In fact, consider the regularization {uδ } of Section 4.2 which clearly satisfies uδ (r, ϕ) = uδ (r) since for any real number ϕ0 the function uδ (r, ϕ + ϕ0 ) is Jδ -minimizing with respect to the boundary data u0 as well. Hence, the uniqueness of minimizers proves the claim for uδ . Now recall the reasoning of Section 4.3.2, Step 3, to see that almost everywhere ∇u∗ = ∇a u∗ = ∇v = ∇a v , where u∗ = u∗ (r) denotes a L1 -cluster point of the sequence {uδ }. Thus we have (58) recalling v ∈ W11 (Ω). Step 2. We next claim that v takes the boundary data u0 in the sense that the trace of v on ∂Ω is just u0 , i.e. v(ρ1 ) = −a

and

v(ρ2 ) = a .

(59)

In order to prove (59) we consider the comparison function ⎧ ⎪ ⎪ ⎨ v(r) − v(ρ1 ) − a ρ1 < r < ρ , w(r) = ⎪ ⎪ ⎩ v(r) − v(ρ2 ) + a ρ ≤ r < ρ2 , and assume by contradiction that (59) is violated. If we observe that 1

f∞ (r, P ) = α k (r) |P | ,

ρ1 < r < ρ2 , P ∈ R2 ,

then we obtain with the notation of Appendix A

f (r, ∇ w) dx + a

K[w] = Ω

d|∇s w| .

(60)

∂Bρ (0)

Here we used the fact that ∇s w is supported on ∂Bρ (0) and that w takes its boundary data u0 on ∂Ω. From [AFP], Theorem 3.77, p. 171, one gets


135

x dH1 . ∇s w∂Bρ (0) = v(ρ1 ) − v(ρ2 ) + 2 a |x| Thus, (60) may be written in the form

f (r, ∇a w) dx + 2 π ρ v(ρ1 ) − v(ρ2 ) + 2 a Ω ≤ f (r, ∇v) dx + 2 π ρ v(ρ1 ) + a + v(ρ2 ) − a .

K[w] =

(61)

Ω

Now choose γ sufficiently large such that for i = 1, 2 ρ 1

ρi α k (ρi )

< 1.

(62)

Then we obtain f (r, ∇v) dx +

K[w] < Ω

2 +

1 k

|u0 − v(ρi )| dH1 = K[v] ,

α (ρi )

i=1

∂Bρi (0)

hence the desired contradiction (recall Theorem A.3, iii), which remains valid with an additional smooth x-dependence). Step 3. Now we make use of the Euler equation for the generalized minimizer v which takes the standard form since v is assumed to be of class W11 (Ω), i.e. we have ∇P f (r, ∇v) · ∇ψ dx = 0 for all ψ ∈ C01 (Ω) . (63) Ω

In particular, this is true for test-functions ψ = ψ(r) ∈ C01 ((ρ1 , ρ2 )). In the following the derivative with respect to r is denoted by “ ˙ ”. Then, again using polar coordinates, ∇v = (cos ϕv, ˙ sin ϕv), ˙ and with the notation ∇P f (r, ∇v) = g(r, |v|)∇v ˙ we obtain from (63)

ρ2

ρ1

2π

g(r, |v|) ˙ v˙ ψ˙ r dr dϕ = 0

for all ψ = ψ(r) ∈ C01 ((ρ1 , ρ2 )) .

0

As a consequence, there is a real number λ ∈ R such that for all r ∈ (ρ1 , ρ2 ) g(r, |v|) ˙ v˙ r = λ . With the representation ⎧ 1 −1 ⎪ ⎪ ⎨ 1 + α(r) |v| ˙ k k α(r) |v| ˙ >ε, ˙ k−2 if |v| g(r, |v|) ˙ = 1 −1 ⎪ 2 ⎪ ⎩ b 1 + α(r) |v| ˙ l−2 + 2 c [α(r)] k if |v| ˙ < ε ˙ l l α(r) |v|

(64)

136


we have to distinguish two cases. Case 1. If |v| ˙ < ε, then using the formulas for b and c we immediately see that g(r, |v|) ˙ ≤ c|ε|k−2 , in particular ε > |v| ˙ =

|λ| ≥ c |λ| g(r, |v|) ˙ r

(65)

for some positive constant c. Case 2. If |v| ˙ > ε, then (64) implies by elementary calculations (note that |λ| > 0 in the case at hand) − k k 1 |v| ˙ −k α− k−1 + α− k−1 = |λ|/r k−1 .

(66)

Observe that, as a consequence of (66),

|λ|/r

k

≤ α(r) .

(67)

Now, again with some simple computations, (66) gives

1 |λ|/r k−1 |v| ˙ = 1 . 1 k k k−1 1 α k α k−1 − |λ|/r

(68)

Summarizing both cases we have the formulas (65) and (68) for |v|. ˙ We then choose λ0 > 0 sufficiently small such that (68) (which is independent of the parameter ε > 0) implies |v| ˙ ≤ 1 and assume that |λ| < λ0 . Then, by (65) and (68) we see that |v| ˙ ≤ 1 for all r ∈ (ρ1 , ρ2 ). On the other hand, v takes its boundary data and v(r) is of class W11 ((ρ1 , ρ2 )), hence v(r) is an absolutely continuous function and we may write 1 ρ2 ρ2 − ρ1 1 a = |v(ρ1 ) − v(ρ2 )| ≤ . (69) |v(r)| ˙ dr ≤ 2 2 ρ1 2 This gives a contradiction if a is sufficiently large and we may assume |λ| ≥ λ0 which was chosen independent of ε. Hence, if Case 1 holds true, then (65) yields (70) ε ≥ c |λ0 | , and we choose ε sufficiently small such that this is not possible. Once it is established that Case 2 holds for all r ∈ (ρ1 , ρ2 ), we obtain from (67) |λ| ≤

inf

r∈(ρ1 ,ρ2 )

1

α k (r) r .

(71)

Moreover, (68) gives the right representation and using (67), (71), α ≥ 1 and k > 2 we estimate


137

2−k

1

α k(k−1) α k(k−1) |v| ˙ ≤ ≤ 1 1 1 1 k k k k k−1 k−1 1 k−1 k−1 k α α α − |λ|/r − |λ|/r $− k1 # k k−1 − 1 1 ≤ α k−1 − r−1 inf [α1/k (r) r] =: h(r) k .

(72)

r∈(ρ1 ,ρ2 )

Here we first note that h(r) is independent of λ, in particular h(r) does not depend on the boundary values u0 given in terms of a. Moreover, h(r) ≥ 0 is evident by definition. Finally, the zeros of h(r) are of finite number and simultaneously (by (62) interior) minima of h(r). With Taylor’s formula we obtain the expansion h(r) ≈ c (r − r0 )2

near the zeros r0 of h(r) ,

since h (r0 ) is not vanishing. In fact, assume by contradiction that h(r0 ) = 0 ,

h (r0 ) = 0 ,

h (r0 ) = 0 .

This leads (by elementary calculations) to k(1 + k)ρk , 2γ

r02+k ≤

(73)

hence if γ = γ(k, ρ1 , ρ2 ) is chosen sufficiently large, then no such zeros are possible for the radii under consideration. With the above expansion we may choose a < ∞ such that ρ2 − 1 h(r) k dr < a . (74) ρ1

This proves the theorem since (72) and (74) contradict (69). Remark 4.41. Let us again concentrate on the regularizing sequence {uδ } with L1 -cluster point u∗ as studied in the previous sections. Then it is not difficult to locate the singular set of u∗ in the situation at hand. To this purpose denote by ρ0,i , i = 1, . . . , M , the minima (of finite number, lying in the interior 1 of (ρ1 , ρ2 ) by (62)) of the function α k (r)r on (ρ1 , ρ2 ). We then have s ∗

spt ∇ u ⊂

M /

∂Bρ0,i (0) .

(75)

i=1

In fact, the sequence of radially symmetric functions {uδ } = {uδ (r)} yields a minimizing sequence of the one-dimensional energy (Ω = I = (ρ1 , ρ2 )) f (r, |w(r)|) ˙ r dr JI [w] := I

138


with respect to W11 (I)-comparison functions w(ρ1 ) = −a, w(ρ2 ) = a. In this sense u∗ provides a generalized JI -minimizer. Now we again extend the ideas of [GMS1] and let u˙ ∗a , u˙ ∗s denote the Lebesgue-decomposition of u˙ ∗ in the absolutely continuous and singular part, respectively. Moreover, Corollary 3.33, [AFP], p. 140, on the decomposition of functions of bounded variation defined on intervals allows us to choose v˜(r) ∈ W11 (I) such that for almost all r ∈ I v˜˙ (r) = u˙ ∗a (r)

and

v˜(ρ1 ) = u∗ (ρ1 ) .

Next let v(r) differ from v˜(r) just by additional jumps at the points ρ0,i such that M 1 + v˙ s (r) = δρ u˙ ∗ , M i=1 0,i I s where δρ0,i denotes the Dirac-measure centered at ρ0,i , i = 1, . . . , M . Note that ∗ u˙ a (t) dt + u˙ ∗s = u∗ (ρ2 ) − u∗ (ρ1 ) , I I M 1 + v˙ a (t) dt + u˙ ∗s δρ = v(ρ2 ) − v(ρ1 ) M i=1 I 0,i I I also implies v(ρ2 ) = u∗ (ρ2 ). Thus we obtain 1 ∗ f (r, |u˙ a (r)|) r dr + α k (r) r d|u˙ ∗s |(r) I I 1 f (r, |v˙ a (r)|) r dr + α k (r) r d|v˙ s |(r) ≤ I I 1 ∗ ≤ f (r, |u˙ a (r)|) r dr + min α k (r) r |u˙ ∗s | ≤ K[u∗ ] r∈(ρ1 ,ρ2 )

I

and our claim (75) is proved.

I

Remark 4.42. Although u∗ as discussed above is not of class W11 (Ω) and although we do not know whether u∗ is of class C 2,α on the complement of the singular set we might conjecture that there exist analogous examples in the case μ = 3 providing W11 -minimizers of f (∇w) dx + f∞ (u0 − w) ν dHn−1 → min , Ω

∂Ω

which are smooth on the complement of a finite number of interior spheres. However, if solutions of this kind exist, then they are due to the non-convexity of Ω. In fact, consider a smooth convex domain Ω, assume that n ≥ 2, N = 1, and suppose that there is a W11 (Ω)-solution which is of class C 2,α near the boundary ∂Ω. Then, on account of the uniqueness of solutions (up to a constant), we apply Hilbert-Haar arguments (compare [MM]) to see that the singular set is empty. In this sense, as the typical behavior, singularities must concentrate near the boundary.


139

Remark 4.43. In order to show rigorously that our regularity theory breaks down if μ > 3 we have to ensure that the energy density f studied in Theorem 4.39 satisfies Assumption 4.22 (of course now with ellipticity exponent μ = k + 1). Here it is clearly sufficient to consider (P ∈ Rn ) 1 f (P ) = 1 + |P |k k , k > 2 , and to study Assumption 4.22 with respect to 1 f˜(x, P ) = f α(x) P , α(x) = 1 + |x|2 k , whenever |P | > 1 and x ∈ B1 (0) ⊂ Rn . To this purpose we first observe that direct calculations yield in the case |P | > 1 1 −2 (76) D2 f (P ) · P = 1 + |P |k k (k − 1) |P |k−2 P and, as a direct consequence, 3 1+k D f (P )(P, U, V ) ≤ c D2 f (P )(U, V ) + c 1 + |P |2 − 2 |U | |V |

(77)

for all U , V ∈ Rn . For the discussion of f˜ we just have to verify iv), v) and vi) of Assumption 4.22, where iv) immediately follows from (76). Now, note that for 1 ≤ γ ≤ n and |P | > 1 ∂γ ∂γ ∇P f˜(x, P ) = ∂γ ∂γ α(x) ∇f (α(x) P ) +2 ∂γ α(x) D2 f (α(x) P ) · P ∂γ α(x) 2 +α(x) D3 f (α(x) P )(P, P ) ∂γ α(x) +α(x) D f (α(x) P ) · P ∂γ ∂γ α(x) =: 2

4 +

Ii .

i=1

Clearly I1 is uniformly bounded and the same follows for I2 and I4 from (76). I3 is estimated with the help of (77), i.e. 1 3 D f (α(x) P )(α(x) P, P ) α(x) − k ≤ cD2 f (α(x) P ) |P | + c 1 + |P |2 2 ≤ c ,

3 D f (α(x) P )(P, P ) ≤

hence we have v). Finally vi) is established by ∂γ DP2 f˜(x, P )(U, V ) = 2 α(x) ∂γ α(x) D2 f (α(x) P )(U, V ) 2 + α(x) D3 f (α(x) P )(P, U, V ) ∂γ α(x) if we once more recall (77).

5 Bounded solutions for convex variational problems with a wide range of anisotropy

Once more the context of linear growth problems is left. However, we keep in mind that in Section 4.2 – in contrast to Chapter 3 – an application of Sobolev’s inequality was replaced by the study of an additional comparison function. This leads to much better results if we impose some uniform boundedness condition on the regularization (see Assumption 4.11). Recall that (up to a certain extent) the limit case μ = 3 is reached and that we cannot expect regular solutions if the ellipticity is given in terms of μ > 3. With these facts, the conjecture for variational problems with non-standard, superlinear growth is evident: if we impose some boundedness condition as above, then, as a formal correspondence, the relation 1 < q < 4 − μ (for anisotropic power growth integrands 1 < q < 2 + p) is expected to be the best possible one inducing (partially) regular solutions. Note that the relevance of the restriction q < 2 + p was already discovered in [ELM2]: given a uniform L∞ -regularization {uδ }, uniform local higher (up to a certain extent) integrability of |∇u| is established in the vector-valued setting (choosing 2 ≤ p). Nevertheless, the full strength of the above stated correspondence could not be established in the paper [BF5] on anisotropic variational integrals with convex hull property: instead of 1 < q < 2 + p (plus the condition q < pn/(n − 2) of Section 3.4 improving higher integrability to partial regularity) the exponents have to be related by 1 < q < p + 2/3. This is caused by an essential difference to the linear growth situation: in Section 4.2 we benefit from the growth rate 1 = q of the main quantity ∇f (Z) : Z under consideration. Given an anisotropic power growth integrand, we just have the lower bound p < q of this quantity. As a consequence, the techniques again have to be changed such that we do not have to rely on the quantity ∇f (Z) : Z. This leads to the study of Choe’s article [Ch], where bounded solutions with respect to integrands which are of non-standard growth and which are of the form f (Z) = g(|Z|2 ) (an assumption both for N > 1 and the scalar situation) are handled up to 1 < q < p + 1. As a third approach, his results depend on an integration by M. Bildhauer: LNM 1818, pp. 141–159, 2003. c Springer-Verlag Berlin Heidelberg 2003

142

5 Bounded solutions: a wide range of anisotropy

parts combined with a Caccioppoli-type inequality which is slightly different from the one given in Lemma 3.19 or Lemma 4.19, i), respectively. In this Chapter we are interested in the question whether Choe’s Ansatz can be improved. The main idea is to combine some of his arguments with the Caccioppoli-type inequalities as used throughout our whole studies. This gives surprisingly strong results in the sense that the above stated conjecture in fact is valid. This time we start with the vector-valued case in Section 5.1. The consideration of the scalar case in Section 5.2 then slightly leaves the main line: in the previous sections on scalar problems the structured vectorial case is included without essential modifications. This well known reasoning is omitted in the following for the sake of simplicity. Here we want to emphasize two other aspects (which similarly can be included in the previous studies): on one hand it is shown that the study of obstacle problems follows more or less the same ideas as given in the unconstrained case provided that we incorporate a suitable cut-off function. As a consequence, it is sufficient to prove a priori estimates on level-sets. This, on the other hand, allows us to admit some kind of degeneracy. Moreover, we can formally include and substantially extend the energy densities covered in [Ch].

5.1 Vector-valued problems Following the existence theory sketched in Section 3.1 (see Theorem 3.3) we fix the unique solution u of the problem ∇f (∇w) dx → min in KF , (P) J[w] := Ω

where F is a N-function having the Δ2 -property and where again c1 F (|Z|) − c2 ≤ f (Z)

for all Z ∈ RnN

(1)

is supposed to hold with some positive constants c1 , c2 . The boundary data u0 are supposed to satisfy (2) u0 ∈ JF ∩ L∞ Ω; RN . Our precise assumptions on the energy density f read as Assumption 5.1. The energy density f : RnN → R is a function of class C 2 (RnN ) and its second derivative is estimated for all Z, Y ∈ RnN by − μ q−2 λ 1 + |Z|2 2 |Y |2 ≤ D2 f (Z)(Y, Y ) ≤ Λ 1 + |Z|2 2 |Y |2 .

(3)

Here λ, Λ denote some positive constants and the exponents μ ∈ R, q > 1, are related by

5.1 Vector-valued problems

q < 4−μ. Moreover, the representation formula f (Z) = g |Z1 |, . . . , |Zn | ,

Z = Z1 , . . . , Zn ∈ RnN ,

143

(4)

(5)

is supposed to be valid for some function g which is increasing with respect to each argument. Finally, we assume the lower bound (1). Remark 5.2. Let us shortly discuss the condition (5). i) It is proved in [BF5] that (5) implies the convex hull property: if Im(u0 ) ⊂ K for a compact convex set K ⊂ RN , then the solution u (together with its regularization) also satisfies Im(u) ⊂ K. In fact, let π: RN → K denote the projection onto K, hence Lip(π) = 1. Then Lemma B.1 of [BF5] proves |∂γ (π ◦ u)| ≤ |∂γ u|, γ = 1, . . . , n. As a result, the comparison function v := π ◦ u is minimizing, g |∂1 v|, . . . , |∂n v| ≤ g |∂1 u|, . . . , |∂n u| , and the uniqueness of solutions proves the claim. ii) Condition (5) can be replaced by any other assumption which gives an appropriate maximum principle (once more compare [DLM]). For instance we could follow the lines of Assumption 4.11 or Remark 4.12. In the situation at hand we prefer the formulation (5) as a natural approach to anisotropic variational problems. iii) With the above remark it is again evident that the expression “bounded solutions” occurring in the heading of this chapter is meant in the sense of “uniformly bounded regularizations”. Let us give some further comments on the choice of f which will be the same in the scalar situation. Remark 5.3. i) In contrast to the notion of (s, μ, q)-growth there is no need to specify the growth rate s of the variational integrand in the following considerations. We just have the bounds induced by (3) (see ii) below). ii) The discussion of Remark 4.2 shows: if μ < 1, then ellipticity is good enough to improve (1) to the power growth estimate (with suitable constants c1 , c2 > 0) c1 |Z|2−μ − c2 ≤ f (Z)

for all Z ∈ RnN .

Moreover, f is at most of growth rate q (recall Remark 3.5, iii)). iii) Note that (4) – in complete accordance with Section 4.2 – implies μ < 3.

144


iv) It should be kept in mind that anisotropic power growth examples (like discussed in Examples 5.7 below) satisfy (4) whenever q < 2+p. Now the main result of this section is formulated: Theorem 5.4. Let u denote the unique solution of the problem (P) where f satisfies Assumption 5.1 and where the boundary values u0 are given according 0 Ω there is a positive to (2). Then for any q < s < 4 − μ and for any Ω number c such that |∇u|s dx ≤ c < ∞ . Ω

The following comments will also remain unchanged in the scalar case. Remark 5.5. i) A corresponding result is true for local minimizers of the energy J[w]. ii) In terms of anisotropic integrands with (p, q)-growth, higher integrability was established in Lemma 3.17 whenever q < p

n+2 , n

(a)

and there was no need to use L∞ -bounds for the solution. Hence, at first glance one may wonder about the case 2+p < p

n+2 , n

(b)

since then the hypothesis q < 2 + p implies (a). Thus, higher integrability holds without an additional boundedness condition. But (b) is equivalent to p > n, hence, by Sobolev’s embedding theorem, boundedness becomes no restriction at all. iii) With ii) it is clear that the results extend to the case n + 2 q < max 4 − μ, (2 − μ) . n iv) We do not want to state a conjecture on the sharpness of our results. Nevertheless, recalling Section 4.4 together with the above mentioned correspondences and having the discussion of Section 3.5 in mind, we at least note that our assumptions are reasonable and consistent. Given Theorem 5.4 we obtain the following corollary on partial regularity. Here we follow the blow-up arguments of Section 3.4.3 which remain unchanged once a Caccioppoli-type inequality and higher local integrability of the gradient are verified. Some more details are outlined in [BF5], let us just note (see Remark 5.8 below) that the way of regularizing the problem is irrelevant since these ingredients are formulated in terms of the solution u.


145

The restriction

n if n ≥ 3 (∗) n−2 is due to the needed properties of the auxiliary functions ψm which were introduced in Section 3.4.3.3. Since our boundedness condition does not improve the Caccioppoli-type inequality, which in return is the basis of the discussion of ψm , we cannot expect to get rid of assumption (∗). q < (2 − μ)

Corollary 5.6. The hypotheses of Theorem 5.4 together with the condition n if n ≥ 3 q < min 4 − μ, (2 − μ) n−2 yield an open set Ω0 ⊂ Ω of full measure, |Ω − Ω0 | = 0, such that u ∈ C 1,α (Ω0 ; RN ) for any 0 < α < 1. Before we are going to prove Theorem 5.4 let some examples illustrate our result. Example 5.7. i) As a first example we consider the anisotropic energy density p q f (Z) = 1 + |Z|2 2 + 1 + |Zn |2 2 ,

Z = (Z1 , . . . , Zn ) ∈ RnN ,

with exponents 2 ≤ p < q. This structure is imposed in [AF4] to obtain partial regularity under a rather weak condition relating p and q (see [BF5] for a detailed comparison with the results of [PS], [BF5] and Section 3.4). ii) If we do not have the above decomposition, for instance (2 ≤ p < q) q p2 , Z = (Z1 , . . . , Zn ) ∈ RnN , f (Z) = 1 + |Z|2 + 1 + |Zn |2 p or if the energy density is completely anisotropic in the sense that f (Z) =

n +

φi (Zi ) ,

qi φi = 1 + |Zi |2 2 ,

Zi ∈ RN ,

i=1

with exponents qi ≥ 2, then the results of [AF4] do not apply any more. However, with the notation introduced in (3) (letting p = 2 − μ), partial regularity follows from [BF5] if q < pn/(n − 2) and q < q0 := max{p + 2/3; p(n + 2)/n}. In the following we will improve p + 2/3 to p + 2. iii) Let us finally discuss an example which is the most interesting one from our point of view. Let Z = (Z1 , Z2 ) ∈ RkN × R(n−k)N , 1 ≤ k < n. Moreover, suppose we are given exponents 1 < p < q < 2 and p q f (Z) = 1 + |Z|2 2 + 1 + |Z2 |2 2 . In this subquadratic case (by elementary calculations) the estimate

146


p−2 λ 1 + |Z|2 2 |Y |2 ≤ D2 f (Z)(Y, Y ) ≤ Λ |Y |2 is the best possible one. As a consequence, no regularity results are available up to now if p is close to 1 – even if (q − p) becomes very small. Hence, with the trivial inequality 2 < p + 2, our theorem really covers a new class of variational integrals. We now come to the Proof of Theorem 5.4 and we assume that all the hypotheses stated there are valid. We denote by (u)ε the ε-mollification of the minimizer u under consideration through a family of smooth mollifiers, we fix a ball B := BR (x0 ) Ω and assume that B ⊂ {x ∈ Ω : dist(x, ∂Ω) > ε} for any small ε > 0 as above. Moreover, we choose some exponent t > max{2, q} and let for any δ ∈ (0, 1) t fδ (Z) := f (Z) + δ 1 + |Z|2 2 ,

Z ∈ RnN .

Then uδ (= uεδ ) is defined as the unique solution of the Dirichlet problem ◦ (Pδ ) fδ (∇w) dx → min , w ∈ (u)ε |B + Wt1 B; RN . Jδ [w, B] := B

Remark 5.8. In Remark 3.13 we summarized the reasons why the regularization of Chapter 3 was done with respect to t = q. The technique outlined below relies on the condition t > max{2, q}: the discussion of asymptotically regular integrands (compare [CE] or [GiaM], Theorem 5.1) includes the vectorial case and yields 1 2 B; RN ∩ W2,loc (6) (B; RN . uδ ∈ W∞,loc If δ = δ(ε) is chosen sufficiently small (recall the conclusion of Chapter 3.3), then the counterpart of Lemma 3.28 reads as Lemma 5.9. With the above notation we have i) ii) iii) iv) v) vi)

uδ(ε) WF1 (B;RN ) ≤ const < ∞; uδ(ε) u in W11 (B; RN ) and almost everywhere as ε → 0; sup |uδ (ε)| ≤ sup |u| < ∞; B BR+ε (x0 ) t 1 + |∇uδ(ε) |2 2 dx → 0 as ε → 0; δ(ε) B f (∇uδ(ε) ) dx → f (∇u) dx as ε → 0; B B fδ(ε) (∇uδ(ε) ) dx → f (∇u) dx. B

B

Keeping Lemma 5.9 in mind, we again use the notation “δ → 0” instead of “ε → 0 and δ(ε) sufficiently small” (compare Lemma 3.28). Finally let us recall the Caccioppoli-type inequality (see Lemma 3.19)


147

Lemma 5.10. There is a real number c > 0, independent of δ, such that for any η ∈ C0∞ (B), 0 ≤ η ≤ 1, D2 fδ (∇uδ )(∂γ ∇uδ , ∂γ ∇uδ ) η 2 dx B 2 D fδ (∇uδ ) |∇uδ |2 |∇η|2 dx . ≤ c B

Remark 5.11. Note that this time the proof is standard: on account of (6) we may differentiate the Euler equation and take ϕ = η 2 ∂γ uδ as an admissible test-function. We obviously (see Lemma 5.9) have established Theorem 5.4 once uniform local higher integrability of the regularization is proved in the sense of Theorem 5.12. With the above stated hypotheses, for any q < s < 4 − μ and for any ball Br (x0 ), r < R, there is a constant c just depending on the data, on supB |(u)ε | and on r and s, such that |∇uδ |s dx ≤ c < ∞ . Br (x0 )

Proof. If s is fixed as above then it is possible to define q + μ − 4 < α := s + μ − 4 < 0 ,

(7)

where the negative sign of α gives 0 < σ := 2 + α −

μ α−μ =: σ . < 2+ 2 2

(8)

Note that we may suppose without loss of generality that |α| is sufficiently small in order to obtain the positive sign of σ. Alternatively, we observe that in the case of a negative sign the second integral on the right-hand side of the inequality (13) below is trivially bounded. By (8) we may choose in addition k ∈ N sufficiently large satisfying 2k

σ < 2k − 2 . σ

Now, given η ∈ C0∞ (B), 0 ≤ η ≤ 1, η ≡ 1 on Br (x0 ), |∇η| ≤ c/(R − r), we introduce the function Γδ = 1 + |∇uδ |2 and recall (6). Hence, uδ is smooth enough to perform the integration by parts α−μ α−μ 2 1+ 2 2k i i 1+ 2 2k dx |∇uδ | Γδ η dx = − uδ · ∇ ∇uδ Γδ η B B 1+ α−μ ≤c |∇2 uδ | Γδ 2 η 2k dx B 3+α−μ Γδ 2 η 2k−1 |∇η| dx . +c B

148


Here we already made use of the fact that uδ is uniformly bounded. If a positive constant M is fixed, then the left-hand side is immediately estimated by α−μ 2+ α−μ 2 1+ 2 2k |∇uδ | Γδ η dx ≥ c Γδ 2 η 2k dx B δ |≥M ] B∩[|∇uα−μ 2+ ≥c Γδ 2 η 2k dx − c(M ) , B

therefore the starting inequality reads as 2+ α−μ 1+ α−μ 2k 2 Γδ η dx ≤ c 1 + |∇2 uδ | Γδ 2 η 2k dx B B 3+α−μ 2

+ B

Γδ

η 2k−1 |∇η| dx

(9)

=: c 1 + I + II . At this point we like to emphasize that the choice (7) of α gives 2+

α−μ s q = > . 2 2 2

(10)

Now for ε > 0 sufficiently small Young’s inequality yields a bound for II II ≤ ε B

≤ε

B

2+ α−μ Γδ 2 2+ α−μ 2

Γδ

η

2k

dx + ε

−1

B

η 2k dx +

−1

−2− α−μ 2

Γδ

cε (R − r)2

B

Γ3+α−μ η 2k−2 |∇η|2 dx δ

1+ α−μ 2

Γδ

(11)

η 2k−2 dx .

Note that the first integral on the right-hand side of (11) may be absorbed on the left-hand side of (9), whereas the second one remains uniformly bounded on account of Remark 5.3, ii), the uniform bound of f (∇uδ ) dx and α < 0. Hence, the theorem is proved if an appropriate estimate for I is found. To this purpose we observe (again ε > 0 is sufficiently small and Young’s inequality is applied) I ≤ ε B

−μ Γδ 2

|∇ uδ | η 2

2 2k+2

dx + ε

−1

B

2+α− μ 2

Γδ

=: ε I1 + ε−1 I2 . Using Lemma 5.9, iv), as well as Lemma 5.10 one obtains

η 2k−2 dx (12)

5.2 Scalar obstacle problems

149

2 D2 fδ (∇uδ )(∂γ ∇uδ , ∂γ ∇uδ ) η k+1 dx

I1 ≤ B

2 D fδ (∇uδ ) |∇uδ |2 η 2k |∇η|2 dx B q t c ≤ Γδ2 η 2k dx + δ Γδ2 η 2k dx (R − r)2 B B q c 2k 2 ≤ 1 + Γ dx . δ η 2 (R − r) B

≤c

As a result, (12) yields (using (10)) cε 2+ α−μ 2+α− μ 2k −1 2 2 + ε 1 + Γ η dx Γ η 2k−2 dx . I ≤ δ δ (R − r)2 B B

(13)

Choosing ε = εˆ(R − r)2 with εˆ > 0 sufficiently small, the first integral on the right-hand side of (13) may also be absorbed on the left-hand side of (9), hence it remains to bound the second one. Here, the negative sign of α and, as a consequence, (8) and our choice of k come into play. For ε˜ > 0 sufficiently small we get with a final application of Young’s inequality −1

εˆ

(R − r)

−2

2+α− μ 2

η 2k−2 dx B α−μ σ 2+ ≤ c εˆ−1 (R − r)−2 ε˜ Γδ 2 η 2k dx + ε˜− σ −σ |B| . Γδ

B

Absorbing terms by letting ε˜ = ε εˆ(R − r)2 , 1 ε > 0, Theorem 5.12 is proved implying the validity of Theorem 5.4 as well.

5.2 Scalar obstacle problems Let us now turn our attention to scalar problems, where we already mentioned in the introduction that now – as a model case – obstacle problems are included whereas we omit the discussion of vector-valued problems with additional structure. The general hypothesis under consideration is slightly different from Assumption 5.1: Assumption 5.13. Let N = 1 and assume that the energy density f : Rn → [0, ∞) satisfies f (0) = 0 and ∇f (0) = 0. Moreover, suppose that f is a strictly convex function of class C 2 (Rn ) which satisfies (1) and the relation (4) of the previous section. Condition (3), Section 5.1, now merely is assumed for all |Z| > 1.

150


Remark 5.14. i) Clearly this setting is much more general than the one considered in [Ch]: we do not suppose f (Z) = g(|Z|2 ) and we just assume the relation q < 4 − μ instead of q < p + 1. As a formal difference, Choe studies energy densities admitting some kind of degeneracy as |Z| → 0. This behavior of the second derivative is covered by Assumption 5.13 since (3), Section 5.1, is not supposed in the case |Z| < 1. We already like to remark that this causes no additional technical difficulties since in any way we have to rely on a suitable cut-off function in order to study obstacle problems. ii) Further comments on Assumption 5.13 were already made in Remark 5.3. The second main result of this chapter is made precise in Theorem 5.15. Fix boundary data u0 according to (2), Section 5.1. Moreover, let the energy density f satisfy Assumption 5.13 and suppose that we are 1 (Ω) such that u0 ≥ ψ almost everywhere. given a function ψ of class W∞,loc Finally, let ψ u ∈ KF := KF ∩ w ∈ KF : w ≥ ψ almost everywhere denote the uniquely determined minimizer of the problem (P) with respect to comparison functions of class Kψ F . Then we have: 1 (Ω). i) u is of class W∞,loc

ii) If in addition the inequality (3), Section 5.1, holds for any Z ∈ Rn then u is of class C 1,α (Ω), if so is the obstacle. Remark 5.16. i) The existence and uniqueness of solutions to scalar obstacle problems follows exactly in the same manner as outlined in Section 3.1. ii) A discussion of (maybe degenerate) obstacle problems is found, for instance, in the papers [MiZ], [CL], [Lin], [MuZ], [Fu1] and [Fu2]. The classical quadratic case is extensively treated in the monographs [KS], [Fri]. Non-standard growth conditions are considered in [Lie1], [BFM]. iii) If we consider degenerate energy densities with (p, q)-growth, then local 1,α -regularity following [BFM] (comLipschitz continuity is improved to Cloc pare [MuZ]). Here an additional hypothesis is needed to control the kind of degeneration of D2 f . iv) Remark 5.5 remains unchanged. In particular from ii), Remark 5.5, we see the substantial improvement of admissible exponents in the case that u0 satisfies (2), Section 5.1 (additionally recall the comparison with known results given in Section 3.5).


151

v) Note that our assumptions are strong enough to obtain a L∞ -bound for u. In fact, we take w := min{u, supΩ u0 } as a comparison function and recall that f (0) = 0 and ∇f (0) = 0. Hence, the strictly convex integrand f attains its minimum in 0 and we get u ≤ supΩ u0 (an analogous estimate holds for the negative sign). Moreover, the same arguments are valid for the regularization introduced below. The Proof of Theorem 5.15 splits into four steps: regularization, linearization and Caccioppoli-type inequalities, uniform local higher integrability and uniform local a priori gradient bounds, where we always assume that the hypotheses of Theorem 5.15 are satisfied. Step 1. (Regularization) Choose (u)ε and B as in the proof of Theorem 5.4 and let (ψ)ε denote the ε-mollification of the obstacle. Then uδ (= uεδ ) is defined as the unique solution of the problem (ψ) fδ (∇w) dx → min , w ∈ Kψ (Pδ ε ) Jδ [w, B] := ε , B

◦

ε 1 where Kψ ε = {w ∈ (u)|B + Wq (B) : (ψ)ε ≤ w almost everywhere} and where we have set q fδ (Z) := f (Z) + δ 1 + |Z|2 2 , Z ∈ Rn .

Remark 5.17. (compare Remark 3.13 and Remark 5.8) As in Chapter 3 the regularization is done with respect to the exponent q. This essentially reduces the difficulties in deriving uniform local a priori gradient bounds. Note that the problem of starting integrability disappears in the scalar case (see Lemma 5.19 below). Remark 5.18. Lemma 5.9 remains valid in the case of obstacle problems, where we just have to observe: the weak W11 - and the almost everywhere convergence of the sequence {uδ } is proved as before. The limit u ˜ respects the obstacle by the almost everywhere convergence, hence lower semicontinuity and the uniqueness of solutions give u ˜ = u. The remaining assertions follow as above. Step 2. (Linearization and Caccioppoli-type inequalities) The study of obstacle problems needs an additional linearization. This procedure is well known, a detailed proof of the following lemma is given in [Fu1]. 2 Lemma 5.19. With the above notation, uδ is of class Wt,loc (B) for any t < ∞ and 1 ∇fδ (∇uδ ) ∈ Wt,loc (B) .

Moreover, the equation

∇fδ (∇uδ ) · ∇ϕ dx =

B

ϕ g dx B

(14)

152


is valid for any ϕ ∈ C01 (B), where g := 1{x∈B: uδ =(ψ)ε }

− div ∇fδ (∇(ψ)ε ) .

Given Lemma 5.19, we now formulate two Caccioppoli-type inequalities on level-sets. Lemma 5.20. Suppose that the hypotheses of Theorem 5.15 are satisfied and fix L > 1 such that for any ε as above L > 1 + ∇(ψ)ε 2L∞ (B;Rn ) . i) Let Bκ := {x ∈ B : Γδ := 1 + |∇uδ |2 > κ}, κ > 1. Then there is a constant c, independent of δ (and ε), such that for any κ > L, for any real number s ≥ 0 and for any η ∈ C0∞ (B), 0 ≤ η ≤ 1, D2 fδ (∇uδ )(∂γ ∇uδ , ∂γ ∇uδ ) Γsδ η 2 dx B2κ 2 D fδ (∇uδ ) Γ1+s |∇η|2 dx . ≤ c δ Bκ

ii) Recall that Γδ := 1 + |∇uδ |2 and let for 0 < r < R A(k, r) = Aδ (k, r) = x ∈ Br (x0 ) : Γδ > k ,

k >1+L.

Then there is a real number c > 0 such that for any η ∈ C0∞ (Br (x0 )), 0 ≤ η ≤ 1 and for any δ ∈ (0, 1)

−μ

Γδ 2 |∇Γδ |2 η 2 dx A(k,r) ≤ c D2 fδ (∇uδ )(∇η, ∇η) (Γδ − k)2 dx . A(k,r)

Proof. ad i). This time we shortly sketch the proof following the idea given in [BFM], Lemma 2.3: fix κ > L and let for all t ∈ R ˜ ˜ −1 t) , h(t) := min max[t − 1, 0], 1 , h(t) = hκ (t) = h(κ (15) i.e. h(t) ≡ 0 if t < κ and h(t) ≡ 1 if t > 2κ. The integrability result of Lemma 5.19 allows us to differentiate the equation (14), thus

2 s D fδ (∇uδ ) ∂γ ∇uδ , ∇ η ∂γ uδ h(Γδ ) Γδ dx B = − g ∂γ η 2 ∂γ uδ h(Γδ ) Γsδ dx . 2

B


153

On the set of coincidence we have almost everywhere ∇uδ = ∇(ψ)ε (see [GT], Lemma 7.7, p. 152), hence the auxiliary function h(Γδ ) vanishes on this set by the choice κ > L. This, together with D2 fδ (∇uδ ) ∂γ uδ ∂γ ∇uδ , ∇Γδ h (Γδ ) Γsδ η 2 dx ≥ 0 , B s D2 fδ (∇uδ ) ∂γ uδ ∂γ ∇uδ , ∇Γδ Γδs−1 h(Γδ ) η 2 dx ≥ 0 B

(which follows from h ≥ 0, s ≥ 0 and 2∂γ uδ ∂γ ∇uδ = ∇Γδ ) yields D2 fδ (∇uδ )(∂γ ∇uδ , ∂γ ∇uδ ) h(Γδ ) Γsδ η 2 dx B ≤ − D2 fδ (∇uδ )(∂γ ∇uδ , ∇η) η ∂γ uδ h(Γδ ) Γsδ dx . B

Finally, Young’s inequality proves the claim after absorbing terms. ad ii). Following the reasoning of Lemma 4.19, ii), we now have to include the side condition. If we are given k > 1 + L then we choose 2 ϕ = η ∂γ uδ max Γδ − k, 0 , η as above. Again Lemma 5.19 shows the validity of the equation (14) and its differentiated version. As before the right-hand side vanishes since k is large enough, thus D2 fδ (∇uδ ) ∂γ ∇uδ , ∂γ ∇uδ (Γδ − k) η 2 dx A(k,r) (16) + D2 fδ (∇uδ )(∂γ ∇uδ , ∇Γδ ∂γ uδ η 2 dx A(k,r) = −2 D2 fδ (∇uδ ) ∂γ ∇uδ , ∇η η ∂γ uδ (Γδ − k) dx . A(k,r)

Here the non-negative first integral on the left-hand side is neglected, the second one satisfies: D2 fδ (∇uδ ) ∂γ ∇uδ , ∇Γδ ∂γ uδ η 2 dx A(k,r) (17) 1 = D2 fδ (∇uδ ) ∇Γδ , ∇Γδ η 2 dx . 2 A(k,r) The right-hand side of (16) is estimated from above by cε D2 fδ (∇uδ ) ∇Γδ , ∇Γδ η 2 dx A(k,r) −1 +c ε D2 fδ (∇uδ )(∇η, ∇η) (Γδ − k)2 dx ,

(18)

A(k,r)

where we made use of Young’s inequality for ε > 0 sufficiently small. Absorbing terms the lemma is proved by (16)–(18) and the ellipticity condition (3), Section 5.1, which can be applied on account of k > 1 + L.

154


Step 3. (Uniform local higher integrability) In the case of scalar obstacle problems higher integrability is improved by an iteration argument to Theorem 5.21. Recall our general assumptions in the case N = 1. Then for any 1 < s < ∞ and for any ball Br (x0 ), r < R, there is a constant c, just depending on the data, supB |(u)ε |, r and s, such that |∇uδ |s dx ≤ c < ∞ . Br (x0 )

Proof. We now fix some non-negative number α ≥ 0 and let β = 4−μ−q > 0, where the positive sign follows from the assumption (4), Section 5.1. As a consequence, the counterpart of (8), Section 5.1, reads as 0 < σ := 2 +

α−μ α−β−μ =: σ . < 2+ 2 2

Again we may choose k ∈ N sufficiently large such that 2k

σ < 2k − 2 . σ

Next we have to make use of the auxiliary function h defined in (15). Here we define h with respect to 2κ, κ > L + 1, L given in Lemma 5.20. Once more, the starting inequality is derived by performing an integration by parts which is admissible on account of Lemma 5.19 α−μ 1+ α−μ 2 1+ 2 2k |∇uδ | Γδ h(Γδ ) η dx = − uδ ∂γ ∂γ uδ Γδ 2 h(Γδ ) η 2k dx B B 1+ α−μ ≤c |∇2 uδ | Γδ 2 h(Γδ ) η 2k dx B 3+α−μ Γδ 2 h(Γδ ) η 2k−1 |∇η| dx +c B 3+α−μ Γδ 2 h (Γδ ) |∇uδ | |∇2 uδ | η 2k dx . +c B

This time it is supposed that r < ρ < ρ ≤ R, η ∈ C0∞ (Bρ (x0 )), η ≡ 1 on Bρ (x0 ), |∇η| ≤ c(ρ − ρ)−1 . A lower bound for the left-hand side of this inequality is given by (compare Section 5.1) 2+ α−μ Γδ 2 η 2k dx − c(κ) , B

on the right-hand side we observe that h (Γδ ) identically vanishes outside the set [2κ ≤ Γδ ≤ 4κ]. As an immediate consequence we have


B

3+α−μ 2

Γδ

h (Γδ ) |∇uδ | |∇ uδ | η 2

2k

1+ α−μ 2

dx ≤ c(κ) B2κ

|∇2 uδ | Γδ

155

η 2k dx ,

where the definition of B2κ is the same as in Lemma 5.20. Since it is also obvious that 1+ α−μ 1+ α−μ 2 2k 2 |∇ uδ | Γδ h(Γδ ) η dx ≤ |∇2 uδ | Γδ 2 η 2k dx , B

B2κ

and since an analogous estimate holds for the remaining integral, we arrive at 2+ α−μ 2

B

1+ α−μ 2

η 2k dx ≤ c 1 +

Γδ

B2κ

|∇2 uδ | Γδ

3+α−μ 2

+ B2κ

Γδ

η 2k dx (19)

η 2k−1 |∇η| dx

=: c 1 + I + II . Now, given ε > 0 sufficiently small, II is handled in the same manner as in Section 5.1 −1 c ε 2+ α−μ 1+ α−μ 2k 2 II ≤ ε Γδ 2 η dx + Γ η 2k−2 dx , (20) δ 2 (ρ − ρ) B B where the first integral can be absorbed on the left-hand side of (19). For the discussion of I we first observe that α+β −μ Γδ 2 Γδ 2 |∇2 uδ |2 η 2k+2 dx I≤ε B2κ 2+ α−β−μ −1 2 +ε Γδ η 2k−2 dx =: ε I1 + ε−1 I2 . B2κ

Here we have to check that I1 can be handled with the help of Lemma 5.20, i): by definition it is clear that α + β ≥ 0. Moreover, the choice of κ shows that we have (3), Section 5.1, on the set Bκ (recall that (3), Section 5.1, now merely is assumed whenever |Z| > 1). One gets

α+β

I1 ≤ c B2κ

≤c

Bκ

D2 fδ (∇uδ )(∂γ ∇uδ , ∂γ ∇uδ ) Γδ 2

η k+1

2

dx

α+β 2 D fδ (∇uδ ) Γ1+ 2 η 2k |∇η|2 dx δ

c ≤ (ρ − ρ)2

B

α+β

q

Γδ 2 Γδ2 η 2k dx .

For the last inequality we recall that the regularization was done with respect to t = q. Finally, the choice of β implies

156


cε I ≤ (ρ − ρ)2

B

2+ α−μ Γδ 2

η

2k

dx + ε

−1

2+ α−β−μ 2

B

Γδ

η 2k−2 dx .

(21)

If again ε = εˆ(ρ − ρ)2 and if εˆ > 0 is sufficiently small, then we argue exactly as in Section 5.1, i.e. the first integral on the right-hand side of (21) is absorbed on the left-hand side of (19), whereas −1

εˆ

−2

(ρ − ρ)

2+ α−β−μ 2

η 2k−2 dx B α−μ σ 2+ ≤ c εˆ−1 (ρ − ρ)−2 ε˜ Γδ 2 η 2k dx + ε˜− σ −σ |B| . Γδ

(22)

B

Following (19)–(22), letting ε˜ = ε εˆ(ρ − ρ)2 , 1 ε > 0 and absorbing terms for a last time we have found a real number c = c(κ, α, ρ − ρ, supB |(u)ε |), independent of δ, such that 2+ α−μ 2

B

Γδ

1+ α−μ 2

η 2k dx ≤ c 1 + B

Γδ

η 2k−2 dx

.

(23)

To start an iteration of (23) let ρm = r + (R − r) 2−m ,

m = 0, 1, 2, . . . ,

as well as αm = 2 m ,

i.e.

αm+1 = 2 + αm ,

m = 0, 1, 2, . . . ,

where for any m as above αm is non-negative, hence admissible in the above calculations. Then we obtain (23) for any m = 0, 1, 2, . . . , with the choices ρ = ρm+1 , ρ = ρm , α = αm , i.e.

α −μ 1+ m+1 1+ αm2−μ 2 Γδ dx ≤ c 1 + Γδ dx . Bρm+1 (x0 )

Bρm (x0 )

An iteration completes the proof since α0 = 0 gives a uniformly bounded right-hand side (once more compare Remark 5.3, ii)). Step 4. (Uniform local a priori gradient bounds) With Theorem 5.21 we can apply a Moser-type iteration (as done in [Ch]) to obtain uniform local a priori gradient bounds. We prefer the modification of DeGiorgi’s technique (similar to Section 4.2.4) which seems to be more convenient in the setting of “bad” ellipticity, moreover, the side condition is easily eliminated. Theorem 5.22. Consider a ball BR0 (x0 ) B where we again suppose that the assumptions of Theorem 5.15 are satisfied. Then there is a local constant c > 0 such that for any δ ∈ (0, 1) ∇uδ L∞ (BR0 /2 ,Rn ) ≤ c .


157

Before proving Theorem 5.22 we establish the generalized version of Lemma 4.29, where the case q = 2 was under consideration. Lemma 5.23. Suppose 0 < r < rˆ < R0 such that BR0 (x0 ) B. Then there is a real number c, independent of r, rˆ, R0 , k and δ, satisfying for any k > 1 + L (L as above) n Γδ − k) n−1 dx A(k,r) (24) n - n . 12 n−1 . 12 n−1 - q−2 μ c 2 2 2 ≤ Γδ − k) dx Γδ Γδ dx , n (ˆ r − r) n−1 A(k,ˆ r) A(k,ˆ r) where the sets A(k, r) = {x ∈ Br (x0 ) : Γδ > k} are introduced in Lemma 5.20. Proof of Lemma 5.23. Given η ∈ C0∞ (Brˆ(x0 )), 0 ≤ η ≤ 1, η ≡ 1 on Br (x0 ), |∇η| ≤ c/(ˆ r − r), we proceed exactly as in Lemma 4.29, hence n n n Γδ − k n−1 dx ≤ c I1n−1 + I2n−1 , A(k,r)

where now the first integral on the right-hand side is estimated from above by (recall 2 − μ ≤ q) n n−1

I1

n . n−1 |∇η| Γδ − k dx

-

=

A(k,ˆ r)

- ≤

q−2 2

|∇η| Γδ 2

A(k,ˆ r)

q−2 2

A(k,ˆ r)

n - . 12 n−1 dx

2−q 2

A(k,ˆ r)

-

c ≤ n (ˆ r − r) n−1

Γδ − k

2

Γδ

Γδ − k

2

Γδ

n - . 12 n−1 dx

n . 12 n−1 dx n . 12 n−1 Γδ dx . μ 2

A(k,ˆ r)

Discussing I2 we recall the choice k > 1 + L. Thus it is possible to refer to Lemma 5.20, ii), with the result n n−1

I2

n . n−1 η |∇Γδ | dx

- = A(k,ˆ r)

- ≤

η |∇Γδ | 2

A(k,ˆ r)

- ≤ c

2

−μ Γδ 2

n - . 12 n−1 dx

A(k,ˆ r)

n . 12 n−1 Γδ dx μ 2

n - . 12 n−1 2 D fδ (∇uδ ) ∇η, ∇η Γδ − k dx

A(k,ˆ r)

c ≤ n (ˆ r − r) n−1

- A(k,ˆ r)

n - . 12 n−1 q−2 2 2 Γδ − k dx Γδ

and the lemma is proved.

A(k,ˆ r)

n . 12 n−1 Γδ dx μ 2

2

A(k,ˆ r)

n . 12 n−1 Γδ dx μ 2

158


Proof of Theorem 5.22. Again we merely have to modify the reasoning of Section 4.2.4. Starting with the left-hand side of (24), we fix a real number s > 1 and observe that H¨ older’s inequality implies

q−2 2

A(k,r)

Γδ

Γδ − k

2

dx =

Γδ − k

n 1 n−1 s

A(k,r)

-

≤

Γδ − k

n n−1

q−2

Γδ 2

Γδ − k

n 1 2− n−1 s

. 1s dx

A(k,r)

- ·

q−2 s 2 s−1

A(k,r)

Γδ

dx

Γδ − k

n 1 s (2− n−1 s ) s−1

. s−1 s .

Theorem 5.21 gives a real number c1 (s, n, BR0 (x0 )), independent of δ, -

s s−1

c1 (s, n, BR0 (x0 )) := sup δ>0

such that

q−2 2

A(k,r)

Γδ

Γδ − k

BR0 (x0 )

2

Γδ

n 1 ( q−2 2 +2− n−1 s )

-

dx ≤ c1

Γδ − k

. s−1 s dx < ∞,

n n−1

. 1s dx .

(25)

A(k,r)

In a similar way one obtains

-

μ 2

A(k,ˆ r)

Γδ dx ≤ c2 (t, μ, BR0 (x0 ))

q−2 2

A(k,ˆ r)

Γδ

. 1t dx ,

(26)

where t > 1 is a fixed second parameter. Combining (24), (25) and (26) it is proved that A(k,r)

q−2 2

Γδ

Γδ − k

2

dx ≤

-

c

q−2 2

Γδ − k Γδ n 1 A(k,ˆ r) (ˆ r − r) n−1 s n 1 1 . 12 n−1 - s t q−2 2 · Γδ dx .

2

n 1 . 12 n−1 s dx

(27)

A(k,ˆ r)

For k > 1 + L and for 0 < r < rˆ as above let q−2 2 Γδ 2 Γδ − k dx , a(k, r) := τ (k, r) := A(k,r)

A(k,r)

q−2

Γδ 2 dx .

As before we obtain (h > k > 1 + L) τ (h, r) ≤

1

c (ˆ r − r)

n 1 n−1 s

(h − k)

n 1 1 n−1 s t

1 n 1 (1+ 1t ) τ (k, rˆ) 2 n−1 s

and the choice of parameters given in Section 4.2.4 completes the proof.


159

Since the data of the obstacle just enter through the constant L, the Proof of Theorem 5.15, i), is an immediate consequence of Lemma 5.9 (compare Remark 5.18). Now that i) is established, the second claim follows from the well known papers [MuZ] and [CL] (compare also [FuM] for details). Remark 5.24. As outlined in the introduction of this chapter, the techniques of Section 5.1 and Section 5.2 provide a suitable approach to superlinear problems. In the case of variational problems with linear growth conditions, we preferred arguments based on the discussion of additional comparison functions. One reason to use different techniques is found in Lemma 4.33: the proof crucially relies on the absence of second derivatives of uδ which is achieved by using a test-function independent of ∇uδ . Starting with an integration by parts (as done in Section 5) we would automatically obtain terms involving ∇2 uδ making the arguments impossible. Nevertheless, the close relationship between linear and superlinear growth problems will become evident in the next Chapter where we will use an appropriate version of Theorem 5.22.

6 Anisotropic linear/superlinear growth in the scalar case

Throughout our studies we changed the point of view several times, where the main line skipped from linear to superlinear growth problems and vice versa. In doing so we could benefit from corresponding examples and techniques. For instance, the study of integrands with (s, μ, q)-growth automatically led to the discussion of energy densities with linear growth and μ-ellipticity. Here a boundedness condition substantially improved the results and the same was shown in the following chapter for superlinear growth problems. Next we propose the study of variational integrands with anisotropic linear/superlinear growth, i.e. of energy densities with anisotropic “plastic/elastic” behavior. This closes the line from Chapter 5 to Chapter 2: on one hand, problems of this type are now within reach since regularity results in the case of variational integrals with a wide range of anisotropy are available by the discussion of Chapter 5. On the other hand, we need a refined study of some elements from duality theory as introduced in Chapter 2. Before going into details let us recall Example 5.7, iii), where we want to restrict to the scalar case N = 1 in this chapter. Writing Z = (Z1 , Z2 ) ∈ Rk × Rn−k for some 1 ≤ k < n, we consider the energy density p q f (Z) = 1 + |Z|2 2 + 1 + |Z2 |2 2 with exponents 1 < p < q < 2. In this subquadratic situation the estimate p−2 λ 1 + |Z|2 2 |Y |2 ≤ D2 f (Z)(Y, Y ) ≤ Λ |Y |2 for all Z, Y ∈ Rn and with positive constants λ, Λ is the best possible one. Hence, there is a quite large difference (independent of q − p) of the upper and the lower growth rate if p is close to one. This is the reason why the example does not fit into the discussion of Section 3.3 (recall Section 3.5 for a comparison with other results). On the other hand, with the results of the previous chapter the case − μ q−2 λ 1 + |Z|2 2 |Y |2 ≤ D2 f (Z)(Y, Y ) ≤ Λ 1 + |Z|2 2 |Y |2 M. Bildhauer: LNM 1818, pp. 161–172, 2003. c Springer-Verlag Berlin Heidelberg 2003

(1)

162


can be handled (if u0 is of class L∞ (Ω)) whenever 1 < q < 4 − μ and f is bounded from below by some N-function. Thus, the idea of this chapter is based on a simple observation: if we take the upper exponent q = 2 in (1), then ellipticity up to μ < 2 is admissible. Moreover, in the case 1 < μ < 2 we have linear growth examples as discussed in Chapter 4 and there is the hope to include anisotropic linear/superlinear growth energy densities in our considerations – at least up to a certain extent. As an interesting application one may think of anisotropic materials somewhere between plastic and elastic behavior. Of course there will be no one-to-one correspondence to the results of Chapter 5 which is due to the missing existence of solutions in the setting considered here. Let us already mention that some of our main examples even do not admit a dual solution – this is caused by an additional anisotropic behavior of the superlinear part (compare Remark 6.6 and Remark 6.9 below). Nevertheless, Theorem 5.22 is strong enough to imply local smoothness and uniqueness (up to a constant) of generalized minimizers. The hypothesis of this chapter reads as Assumption 6.1. The energy density f : Rn → R is of class C 2 (Rn ) and admits the decomposition f (Z) = f1 (Z1 ) + f2 (Z2 ) ,

Z = (Z1 , Z2 ) ∈ Rk × Rn−k ,

(2)

for some k ∈ N, 1 ≤ k < n. Here f1 ∈ C 2 (Rk ) is a function of linear growth such that (3) |∇f1 (Z1 )| ≤ A holds for any Z1 ∈ Rk with some constant A. The function f2 ∈ C 2 (Rn−k ) is supposed to satisfy for some 1 < p ≤ 2 c1 |Z2 |p − c2 ≤ f2 (Z2 ) ,

(4)

now for any Z2 ∈ Rn−k and with constants c1 , c2 . Our assumption on the second derivative of f = f1 + f2 is given by − μ λ 1 + |Z|2 2 |Y |2 ≤ D2 f (Z)(Y, Y ) ≤ Λ |Y |2

(5)

for all Z, Y ∈ Rn with constants λ, Λ and with an exponent of ellipticity 1 < μ < 2.

(6)

Remark 6.2. Let us recall some consequences (compare Remark 3.5, iv), and Remark 4.2 i), ii)). i) Whenever it is needed, we will assume without loss of generality that fi (0) = 0, i = 1, 2, and ∇f (0) = 0.


163

ii) The ellipticity condition (5) shows that ∇f (Z) · Z is at least of linear growth which implies a|Z| − b ≤ f (Z)

for all Z ∈ Rn

and with some real numbers a > 0, b. iii) From the right-hand side of (5) we see that f is at most of quadratic growth and therefore (compare [Da], Lemma 2.2, p. 156) |∇f (Z)| ≤ c (1 + |Z|)

for all Z ∈ Rn ,

where c is another positive constant. At this point we should give some examples to describe the class of energy densities which we have in mind by introducing Assumption 6.1. Example 6.3. With the notation of Assumption 6.1 we may take: i) a linear growth integrand with μ-ellipticity, 1 < μ < 2, as given in Example 3.9: let r s − μ 1 + t2 2 dt ds , r ∈ R+ ϕ(r) = 0 , 0

0

and choose f1 (Z1 ) = ϕ(|Z1 |). ii) The most elementary superlinear part is of power growth, i.e. p f2 (Z2 ) = 1 + |Z2 |2 2 ,

1 γ for any x ∈ spt λ. If |t| is sufficiently small, then the same is true if we replace σ by σt := σ + tλ and γ by γ/2. Then (14) implies

A.3 Two uniqueness results

183

div σt (u0 − u) dx + σt : ∇u0 dx spt λ ∗ ∗ ≤ f (σt ) − f (σ) dx + σ : ∇u0 dx .

spt λ

spt λ

spt λ

If we observe that (div σt ) u0 dx + σt : ∇u0 dx − σ : ∇u0 dx spt λ spt λ spt λ = t (div λ) u0 dx + t λ : ∇u0 dx = 0 , spt λ

then we obtain

spt λ

−

t (div λ) u dx ≤ spt λ

f ∗ (σt ) − f ∗ (σ) dx .

spt λ

Dividing through t > 0 and passing to the limit t → 0 we get − (div λ) u dx ≤ ∇f ∗ (σ) : λ dx , spt λ

spt λ

i.e., by definition, the first weak derivative of u is given by ∇f ∗ (σ). Now we are going to prove Theorem A.11. Suppose that the variational integrand f satisfies Assumption A.1 and assume that there exists u∗ ∈ M . Then we have i) The elements of M are solutions of problem (P ) and vice versa. ii) The set M is uniquely determined up to constants. Proof. ad i). On account of the K-minimizing property of u∗ ∈ M and since ∇s u∗ ≡ 0, the representation of K clearly implies that u∗ ∈ M is a solution of (P ). Conversely, consider a solution v ∗ of the problem (P ) and ◦

a J-minimizing sequence {um } from u0 + W11 (Ω; RN ). The minimality of v ∗ gives ∗ ∗ ∗ n−1 f (∇v ) dx + f∞ (u0 − v ) ⊗ ν dH ≤ f (∇um ) dx , K[v ] = Ω

∂Ω

Ω

and i) follows from Theorem A.3, ii) & iii). ad ii). To prove the uniqueness up to a constant, we just observe that f∞ is convex, whereas f is strictly convex. This immediately gives ∇u∗ = ∇u∗∗ almost everywhere for any two generalized minimizers u∗ , u∗∗ ∈ M , hence Theorem A.11.

B Some density results

B.1 Approximations in BV The standard approximation procedure for functions with bounded variation is given in [AG1] (we refer to [Giu2] Theorem 1.17, p. 14, in particular we also like to mention Remark 2.12, [Giu2], p. 38, on the traces of the approximating sequence), where a sequence of smooth, L1 -converging functions is constructed such that in addition the total variations converge as well. In order to obtain the continuity of the relaxed functional Jˆ (compare Proposition 2.22) we need a slight modification which, on one hand, is well known (see [AG2], Proposition 2.3). On the other hand, a rigorous proof is hardly found in the literature. Hence, for the sake of completeness, we first show Lemma B.1. Let w ∈ BV (Ω; RN ). Then there is a sequence {wm } in C ∞ (Ω; RN ) satisfying lim

m→∞

Ω

lim

m→∞

|wm − w| dx

=0; 2 1 + |∇wm | dx = 1 + |∇w|2 .

Ω

Ω

Moreover, the trace of each wm on ∂Ω coincides with the trace of w. Proof. To introduce a precise notation which is compatible with [DT] and [Te], respectively, we let (1) g(Z) = 1 + |Z|2 − 1 on RnN , the conjugate function is given for any Q ∈ RnN by ⎧ ⎨ +∞ if |Q| > 1 , ∗ g (Q) = ⎩ 1 − 1 − |Q|2 if |Q| ≤ 1 .


(2)

186


Then, for any bounded open Lipschitz domain U ⊂ Rn and for any w ∈ BV (U ; RN ), we take as a definition (compare [DT], Proposition 1.2)

∗ − g(∇w) := sup w div κ dx − g (κ) dx . (3) κ∈C0∞ (U ;RnN ), |κ|≤1

U

U

U

Now we follow [Giu2], pp. 14, fix ε > 0 and w ∈ BV (Ω; RN ). Moreover, for any l ∈ N we let 1 l Ωk = Ωk := x ∈ Ω : dist (x, ∂Ω) > , k = 0, 1, 2, . . . . l+k Here l may be chosen sufficiently large such that |∇w| < ε .

(4)

Ω−Ω0

Given Ωk as above, sets Ai are defined by induction: A1 := Ω2 and Ai := Ωi+1 − Ωi−1 ,

i = 2, 3, . . . .

With respect to these Ai we then consider a partition of the unity {ϕi }, i.e. for any i ∈ N ∞ + ∞ ϕi ∈ C0 (Ai ) , 0 ≤ ϕi ≤ 1 , ϕi = 1 . i=1

Finally, η denotes a smoothing kernel and we choose for any i ∈ N a positive number εi sufficiently small such that (letting Ω−1 = ∅)

Ω

Ω

spt ηεi ∗ (ϕi w) ⊂ Ωi+2 − Ωi−2 ; ηε ∗ (ϕi w) − ϕi w dx < 2−i ε ; i

(5)

ηε ∗ (w ⊗ ∇ϕi ) − w ⊗ ∇ϕi dx < 2−i ε . i

Now assume that ε =

1 m

and let wm =

∞ +

ηεi ∗ (ϕi w) .

i=1

Note that here and in the following each sum under consideration is locally finite. The first claim of the lemma is immediate since {wm } is constructed as a smooth sequence and since we have recalling (5) ∞ + ηε ∗ (ϕi w) − ϕi w| dx ≤ ε . |wm − w| dx ≤ i Ω

i=1

Ω

B.1 Approximations in BV

This, together with lower semicontinuity, also proves g(∇w) ≤ lim inf g(∇wm ) dx . m→∞

Ω

187

(6)

Ω

To establish the opposite inequality we fix κ ∈ C0∞ (Ω; RnN ), |κ| ≤ 1. Then, 1∞ by (4), (5), on account of i=1 ∇ϕi ≡ 0 and since the intersection of more than three Ai is empty, we obtain wm div κ dx − g ∗ (κ) dx − Ω Ω ∞ + = − w div ϕ1 ηε1 ∗ κ dx − w div ϕi ηεi ∗ κ dx +

Ω ∞ +

Ω

i=2

Ω

κ : ηεi ∗ (w ⊗ ∇ϕi ) − w ⊗ ∇ϕi dx −

i=1 w div ϕ1 ηε1 ∗ κ dx − g ∗ (κ) dx + 4 ε . ≤ − Ω

(7) g ∗ (κ) dx

Ω

Ω

If we identify κ with its zero-extension to Rn , then we may write ∗ − g (κ) dx = − g ∗ (κ) dx n Ω R =− g ∗ (ηε1 ∗ κ) − g ∗ (κ) dx g ∗ (ηε1 ∗ κ) dx + Rn

Rn

= I + II .

(8)

An estimate for I follows from 0 ≤ ϕ1 ≤ 1, the convexity of g ∗ , g ∗ (0) = 0 and g ∗ ≥ 0: ∗ I ≤ − g (ϕ1 ηε1 ∗ κ) dx = − g ∗ (ϕ1 ηε1 ∗ κ) dx . (9) Rn

Ω

To handle the second integral of (8) we use Jensen’s inequality g ∗ (ηε1 ∗ κ) ≤ ηε1 ∗ g ∗ (κ) . This, together with standard calculations, gives ηε1 ∗ g ∗ (κ) − g ∗ (κ) dx = 0 . II ≤

(10)

Rn

As a result of (7)–(10) it is proved that wm div κ dx − g ∗ (κ) dx − Ω Ω w div ϕ1 ηε1 ∗ κ dx − g ∗ (ϕ1 ηε1 ∗ κ) dx + 4 ε . ≤ − Ω

Ω

(11)

188


Now it is obvious that the comparison function on the right-hand side of (11) is admissible and the opposite inequality to (6) is established: g(∇wm ) dx ≤ g(∇w) . lim sup m→∞

Ω

Ω

Moreover, the approximative functions wm are defined in the same manner as in [Giu2], pp. 14, hence Remark 2.12 of [Giu2], p. 38, remains valid which proves that the trace of each wm on ∂Ω coincides with the trace of w. As a generalization of Lemma B.1 we now choose an approximative sequence with arbitrary prescribed traces. To this purpose we fix again a ˆ Ω and extend our boundary values u0 ∈ bounded Lipschitz domain Ω ◦ ˆ RN ). W 1 (Ω; RN ) to a function of class W 1 (Ω; 1

1

ˆ u0 as above let w ∈ BV (Ω; RN ) and denote by w ˆ Lemma B.2. Given Ω, ˆ the extension via u0 to the domain Ω. Then there exists a sequence {wm } in ˆ u0 |Ω + C0∞ (Ω; RN ) such that if m → ∞ (and if we extend wm by u0 to Ω) i) ii)

ˆ RN ); ˆ in L1 (Ω; wm → w 1 + |∇wm |2 dx → 1 + |∇w| ˆ 2. ˆ Ω

ˆ Ω

Proof. Let us recall the general setting (1)–(3) and fix w, w ˆ as given in the lemma. Now we use a construction as outlined in [Alt], p. 170. Since ∂Ω is Lipschitz, we can cover ∂Ω by open sets V1 , . . . , Vr such that after rotation and translation Vj takes the form n Vj = x ∈ R : (x1 , . . . , xn−1 ) < rj , xn − gj (x1 , . . . , xn−1 ) < hj , where gj is a Lipschitz function. Moreover, we have 0 = xn − gj (x1 , . . . , xn−1 )

⇒ x ∈ ∂Ω ;

0 < xn − gj (x1 , . . . , xn−1 ) < hj

⇒x∈Ω;

0 > xn − gj (x1 , . . . , xn−1 ) > −hj ⇒ x ∈ Ω . Let Vr+1 , Vr+2 denote open sets such that V r+1 ⊂ Ω, V r+2 ⊂ Rn − Ω and ˆ⊂ Ω

r+2 /

Vj .

j=1

In addition, the sets Vj may be chosen such that for any j = 1, . . . , r + 2 g(∇w)(∂V ˆ j) = 0 .

B.1 Approximations in BV

189

ˆ (again following [DT], see also the proof Here we let for any Borel set B ⊂ Ω of Theorem A.6) g(∇w) ˆ

g(∇w)(B) ˆ =

B

=

sup ˆ nN ), |κ|≤1 κ∈C0∞ (Ω;R

ˆ Ω

1B κ∇w ˆ−

g ∗ (κ) dx .

ˆ Ω

Now fix ε > 0 and decrease Vj if necessary (to be a little bit more precise: decrease radius and height of Vj a little bit and obtain sets Vj,ε Vj . Then let U1 = V1 , U2 = V2 − V 1,ε and inductively define Uj , j = 1, . . . , r + 2.). As a result we obtain smooth sets Uj which can be arranged in addition such that r+2 +

ˆ +ε, g(∇w)(U ˆ j ) ≤ g(∇w)( ˆ Ω)

(12)

j=1

g(∇w)(∂U ˆ j) = 0 ,

j = 1, . . . , r + 2 .

(13)

◦

ˆ RN ), hence we may extend ∇w ˆ Suborˆ by 0 to Rn − Ω. Note that u0 ∈W11 (Ω; dinate to the covering {Uj } let {ϕj } denote a partition of the unity, i.e. ϕj ∈

C0∞ (Uj )

0 ≤ ϕj ≤ 1 ,

,

r+2 +

ˆ. ϕj ≡ 1 on Ω

j=0

For each fixed index 1 ≤ j ≤ r and for δ 1 we then define (neglecting with a slight abuse of notation rotations and translations) vˆjδ,ε (x) := ϕj (x) (w ˆ − u0 ) ◦ (x − δ en ) + u0 (x) . δ,ε δ,ε Finally, we let vˆr+1 := ϕr+1 w, vˆr+2 := ϕr+2 u0 and δ,ε

vˆ

:=

r+2 +

vˆjδ,ε .

j=1

Clearly vˆδ,ε − u0 is compactly supported in Ω. Moreover, for δ = δ(ε) sufficiently small we have δ(ε),ε vˆ −w ˆ dx ≤ ε , ˆ Ω

and again lower semicontinuity gives for vˆε = vˆδ(ε),ε g(∇w) ˆ ≤ lim inf g(∇ˆ v ε ) dx . ˆ Ω

ε→0

(14)

ˆ Ω

ˆ RnN ), To prove the opposite inequality we recall the definition (3), fix κ ∈ C0∞ (Ω; |κ| ≤ 1, and observe

190


−

vˆ div κ dx = − ε

ˆ Ω

r + j=1

−

Uj

r+2 +

j=r+1

+

−

dx κ : ∇ ϕj u0 (x) − u0 x − δ(ε) en

Uj

r + j=1

ϕj (x) w(x) ˆ div κ dx

Uj

r + j=1

≤−

ϕj (x) w ˆ x − δ(ε) en div κ dx

ϕj (x) w ˆ x − δ(ε) en div κ dx

Uj

r+2 +

j=r+1

ϕj (x) w(x) ˆ div κ dx + ε ,

Uj

where δ(ε) is assumed to be sufficiently small. Note that this choice does not depend on κ. Now, the right-hand side is estimated by r.h.s. = −

r + j=1

+

Uj

r + j=1

−

r+2 +

+

r+2 +

w(x) ˆ div (ϕj κ) dx

Uj

w(x) ˆ ⊗ ∇ϕj : κ dx

Uj

r + j=1

j=r+1

w(x) ˆ ⊗ ∇ϕj : κ dx + ε

Uj

w ˆ x − δ(ε) en div (ϕj κ) dx

r+2 +

j=1

r+2 +

Uj

j=r+1

+

w(x) ˆ div (ϕj κ) dx +

Uj

r + j=1

−

w ˆ x − δ(ε) en ⊗ ∇ϕj : κ dx

Uj

j=r+1

≤−

w ˆ x − δ(ε) en div (ϕj κ) dx

w(x) ˆ −w ˆ x − δ(ε) en |∇ϕj | dx + ε .

Uj

Once more δ(ε) is decreased in order to bound the last sum by ε. Moreover, 1r+2 j=1 ∇ϕj ≡ 0 implies the third sum to vanish identically. Next observe that g ∗ (ϕj κ) ≤ ϕj g ∗ (κ) follows from the convexity of g ∗ ˆ and from g ∗ (0) = 0, thus we arrive at (κ ≡ 0 on Rn − Ω)

B.2 A density result for U ∩ L(c)

−

191

vˆ div κ dx − g ∗ (κ) dx ˆ ˆ Ω Ω r + − ≤ w ˆ x − δ(ε) en div (ϕj κ) dx − ε

Uj

j=1

+

r+2 +

g ∗ (ϕj κ) dx

Uj

−

w(x) ˆ div (ϕj κ) dx −

Uj

i=r+1

(15)

g ∗ (ϕj κ) dx

+ 2ε .

Uj

˜j Uj , j = 1, . . . , r such that Finally, we choose open sets U r +

˜j ) ≤ g(∇w)( ˆ U

j=1

r +

g(∇w)(U ˆ j) + ε .

(16)

j=1

Note that sets of this kind can be found on account of (13). Moreover, if δ(ε) 1, then we remark that for j = 1, . . . , r −

w ˆ x − δ(ε) en div (ϕj κ) dx − Uj ≤

sup ∞ ˜ nN ), |κ|≤1 κ∈C ˜ ˜ 0 (Uj ;R

−

˜j U

g ∗ (ϕj κ) dx

Uj

w ˆ div κ ˜ dx −

˜j U

g ∗ (κ) ˜ dx

(17) .

Putting together the inequalities (15)–(17) it is proved that −

vˆ div κ dx − ε

ˆ Ω

≤

r +

ˆ Ω

˜j ) + g(∇w)( ˆ U

j=1

≤

g ∗ (κ) dx

r+2 +

r+2 +

g(∇w)(U ˆ j) + 2ε

(18)

j=r+1

g(∇w)(U ˆ j) + 3ε .

j=r+1

Once (18) is established, (12) shows the opposite inequality to (14). Summarizing the results we have proved up to now: there exists a sequence {ˆ vε } such that vˆε − u0 is compactly supported in Ω and such that the convergences claimed in the lemma hold for this sequence. In a last step it remains to apply the standard smoothing procedure (see Lemma B.1) and Lemma B.2 is proved.

B.2 A density result for U ∩ L(c) Here we are going to establish a density result from [BF4] which was needed for the proof of the identification Theorem A.6. With the notation U and L(c) as introduced in Appendix A.2, a precise formulation of this lemma reads as

192


Lemma B.3. Suppose that κ ∈ U satisfies κ(x) ∈ L(c) for some c ∈ R. Then a sequence {κm }, κm ∈ C ∞ (Ω; RnN ) exists such that i) κm → κ in Lt Ω; RnN for any t < ∞ and we have almost everywhere convergence; ii)

div κm → div κ in Ln Ω; RN ;

iii)

∗ κm κ in L∞ Ω; RnN ;

iv)

κm (x) ∈ L(c) for all x ∈ Ω and for any m ∈ N.

Proof. As in the proof of Lemma B.2, the boundary of the Lipschitz domain Ω is covered with sets Vj , j = 1, . . . , r, such that we have the properties stated there. Let V0 denote an open set satisfying V 0 ⊂ Ω, Ω⊂

r /

Vj ,

j=0

and consider a corresponding partition of the unity {ϕj }, i.e. ϕj ∈

C0∞ (Vj )

,

0 ≤ ϕj ≤ 1 ,

r +

ϕj ≡ 1 on Ω .

j=0

For each fixed index j ≥ 1 we let for δ 1 ⎧ ⎨ κ(x + δ e ) ϕ (x) if x ∈ Ω ∩ V ; n j j δ κj (x) := ⎩ 0 if x ∈ Ω − Vj . Note that κjδ ≡ 0 near the “upper” boundary part of Vj , the same is true near the “vertical” boundary parts which follows from the support properties of ϕj and from an appropriate choice of δ. If ω denotes a smoothing kernel, we let r + δ,ρ δ κj (x) , x ∈ Ω . κ (x) := ωρ ∗ ϕ0 κ + j=1

Assuming again the standard representation of the neighborhood Vj we get δ ωρ (y − x) κ(y + δ en ) ϕj (y) dy , ωρ ∗ κj (x) = Rn

and for ρ small enough depending on δ we see that for y ∈ Bρ (x), x ∈ Ω, the point y + δ en belongs to Ω, and ωρ ∗ κjδ (x) is well defined. Clearly ωρ ∗ κjδ ∈ C ∞ (Ω; RnN ) and ρ↓0 ωρ ∗ κjδ → κjδ in Lp Ω; RnN

B.2 A density result for U ∩ L(c)

193

for any p < ∞, moreover (see [Alt], Lemma 1.16, p. 18) in Lp Ω; RnN ,

δ↓0

κjδ → κ ϕj

again for any p < ∞. We further have for x ∈ Ω

div ωρ ∗

κjδ

(x) = −

∂α ωρ (y − x) κα (y + δ en ) ϕj (y) dy

Bρ (x)

and since it is sufficient to consider x ∈ Ω ∩ Vj , we see that the arguments on the right-hand side are compactly supported in Ω. Moreover, y → ωρ (y − x) has compact support in Bρ (x), thus

div ωρ ∗

κjδ

ωρ (y − x) div κ(y + δ en ) ϕj (y)

(x) = Bρ (x)

+κ(y + δ en ) ∇ϕj (y) dy

and, as above, ρ↓0 div ωρ ∗ κjδ ) → div κ(· + δ en ) ϕj + κ(· + δ en ) ∇ϕj in Ln (Ω; RN ). The right-hand side converges to div κ ϕj + κ ∇ϕj in Ln (Ω; RN ) as δ ↓ 0. So, if we first fix a sequence δm ↓ 0, we find a sequence {ρm } depending on {δm } such that the convergence properties i) and ii) hold ∗ ˜ for κ m := κ δm ,ρm . The boundedness of κ m L∞ (Ω;RnN ) implies κ m κ ∞ nN ∞ nN in L (Ω; R ) for a subsequence and some tensor κ ˜ ∈ L (Ω; R ), but i) shows κ ˜ = κ. It remains to prove iv). Jensen’s inequality applied to the measure ωρ (x − ·)Ln gives f

∗

κ

δ,ρ

(x) ≤

ωρ (x − y) f

∗

ϕ0 κ +

Bρ (x)

r +

κjδ (y) dy

j=1

and if we recall the definition of κjδ we see that f ∗ is evaluated on the convex combination r + ϕj (y) κ(. . . ) , ϕ0 (y) κ(y) + j=1

where κ(. . . ) has an obvious meaning for j = 1, . . . , r. Our assumption κ ∈ L(c) almost everywhere then implies f

∗

ϕ0 κ +

r + j=1

κjδ

(y) ≤ c ,

194


i.e. κ δ,ρ ∈ L(c). For technical reasons (see the proof of Theorem A.6), we also need the following Remark B.4. Recall the definition of κjδ (x) given in the proof of Lemma B.3. Clearly this definition makes sense for points x such that x + δ en ∈ Ω, i.e. −δ ≤ xn − gj (x1 , . . . , xn−1 ) , so that (combined with the smoothing procedure outlined above) κjδ ∈ C ∞ (Vj ∩ [−δ/2 ≤ xn − gj (x1 , . . . , xn−1 )]). If we then let ⎧ ⎨ 1 if x ∈ V , x − g (x , . . . , x j n j 1 n−1 ) ≥ 0 , ψjδ (x) := ⎩ 0 if x ∈ V , x − g (x , . . . , x ) ≤ −δ/4 , j

n

j

1

n−1

ψjδ ∈ C0∞ (Rn ), 0 ≤ ψ ≤ 1, then the function ψjδ κjδ , j = 1, . . . , r, is of class ˆ RnN ), where Ω ˆ Ω is some bounded Lipschitz domain and δ is chosen C0∞ (Ω; ˆ is fixed as above, then the sequence {κm } sufficiently small. As a result, if Ω ˆ RnN ). given in Lemma B.3 may be in addition assumed to be of class C0∞ (Ω; Moreover, again by the convexity of f ∗ (further recall that f ∗ (0) = 0), the ˆ level set property continues to hold on the extended domain Ω.

B.3 Local comparison functions A helpful tool which was used in Sections 2.3, 2.4 and 4.3 is the construction of local comparison functions as given in [BF1]. With the notation ˆ ˆ f (∇w) dx for any open set Ω J[w; Ω] = ˆ Ω

we now prove for f given as in Assumption 2.1 Lemma B.5. Consider a sequence {um } ⊂ W11 (Ω; RN ) such that: i) um → u∗ in L1 (Ω; RN ) as m → ∞; ii) sup um W11 (Ω;RN ) < ∞. m∈N

Then we can find a subsequence (still denoted by {um }) with the following properties: for any x0 ∈ Ω and for almost any ball BR (x0 ), B2R (x0 ) Ω, there is a sequence {wm } ⊂ W11 (Ω; RN ) such that i) wm → u∗ in L1 (Ω; RN ) as m → ∞; ii) lim sup J wm ; BR (x0 ) ≤ lim inf J um ; BR (x0 ) ; m→∞

m→∞

B.3 Local comparison functions

195

iii) lim sup J[wm ; Ω] ≤ lim inf J[um ; Ω]; m→∞

m→∞

iv) wm |∂BR (x0 ) = u∗|∂BR (x0 ) , where the traces are well defined functions of class L1 (∂BR (x0 ); RN ); v) wm |Ω−B2R (x0 ) = umk |Ω−B2R (x0 ) for any m ∈ N, in particular the boundary values on ∂Ω are preserved; vi) wm |BR/2 (x0 ) = uml |BR/2 (x0 ) for any m ∈ N. Here {umk } and {uml } denote some appropriate subsequences of {um }. Proof. We have u∗ ∈ BV and we may also assume that ∇um ∇u∗ ,

|∇um | μ

as m → ∞

in the sense of measures where μ denotes a Radon measure of finite mass. We may choose a radius R > 0 according to B2R (x0 ) Ω

and

μ(∂BR (x0 )) = 0 = |∇u∗ |(∂BR (x0 )) .

(19)

This implies (see [Giu2], Remark 2.13) that u∗|∂BR (x0 ) is well defined. With Tε := {x : R − ε < |x − x0 | < R + ε}, ε > 0 sufficiently small, we further obtain:

(20) lim lim sup |∇um | dx = 0 , ε→0 m→∞ Tε lim |∇u∗ | = 0 . (21) ε→0

Tε

For (21) we just observe using (19) 2 ε→0 |∇u∗ | −→ |∇u∗ | Tδ = |∇u∗ |(∂BR (x0 )) = 0 . Tε

δ>0

Next, let ϕε ∈ C0 (T2ε , [0, 1]), ϕε = 1 on Tε . Then

Tε

hence

|∇um | dx ≤

m→∞

ϕε |∇um | dx −→ T2ε

ϕε dμ , T2ε

ε→0

|∇um | dx ≤ μ(T2ε ) −→ μ(∂BR (x0 )) = 0 ,

lim sup m→∞

Tε

thus (20) holds. With R fixed we now let 1 N ∗ K := w ∈ W1 BR (x0 ); R : w|∂BR (x0 ) = u|BR (x0 ) .

196


Note that K = ∅ on account of [Giu2], Remark 2.12 and [Giu2], Theorem 2.16. We first claim that there exists a sequence {vk } ⊂ K for which conclusion (ii.) holds: lim sup J vk ; BR (x0 ) ≤ lim inf J um ; BR (x0 ) . (22) m→∞

k→∞

For proving (22) consider u ˜m ∈ C ∞ (BR (x0 ), RN ) with (compare again [Giu2], Remark 2.12) ∗ |˜ um − u | dx → 0 , |∇˜ um | dx → |∇u∗ | BR (x0 )

BR (x0 )

BR (x0 )

as m → ∞ and such that u ˜m|∂BR (x0 ) = u∗|∂BR (x0 ) . Then lim

ε→0

|∇˜ um | dx

lim sup m→∞

= 0.

(23)

BR (x0 )∩Tε

In fact, let Aε := {x : R − ε < |x − x0 | < R}. We have (see [Giu2], Prop. 1.13)

lim sup m→∞

BR (x0 )∩Tε

|∇˜ um | dx = lim sup m→∞ ≤

|∇˜ um | dx ∗ |∇u | ≤ |∇u∗ | → 0

BR (x0 )∩Aε

BR (x0 )∩Aε

T2ε

as ε → 0 and (23) follows. We define ⎧ ⎨ on BR (x0 ) − Tε , um ε vm := ⎩ u + ε−1 (˜ um − um )(|x| − R + ε) on Aε = BR (x0 ) ∩ Tε . m ε ε Both parts of vm induce the same trace on ∂BR−ε (x0 ), thus vm is of class 1 N ε W1 (BR (x0 );R ) and in addition vm ∈ K. Since f is of linear growth, the ε ε ) dx follows from the discussion of Aε |∇vm | dx: behavior of Aε f (∇vm

Aε

ε |∇vm |

1 dx ≤ 2 |∇um | dx + |∇˜ um | dx + ε Aε Aε

|um − u ˜m | dx . BR (x0 )

According to (20) and (23) it is possible to define a sequence εk → 0 such that 1 1 and lim sup |∇um | dx ≤ |∇˜ um | dx ≤ lim sup k k m→∞ m→∞ Aε Aε k

k

for all k ∈ N, thus, for any k ∈ N, there is mk ∈ N such that 2 2 and |∇um | dx ≤ |∇˜ um | dx ≤ k k Aε Aε k

k

B.3 Local comparison functions

197

for all m ≥ mk . Recalling the L1 –convergences um , u ˜m → u∗ on BR (x0 ), we assume in addition 1 |um − u ˜m | dx ≤ εk εk BR (x0 ) for all m ≥ mk , k ∈ N. Putting together these estimates we get ε k f (∇um ) dx + αk , m ≥ mk , k ∈ N , J vm ; BR (x0 ) ≤ BR (x0 )−Tεk

for a sequence αk ≥ 0, αk → 0 as k → ∞. By enlarging mk , if necessary, we can arrange f (∇umk ) dx = lim inf f (∇ul ) dx . lim k→∞

l→∞

BR (x0 )

εk Finally vk := vm is introduced. Then k

lim sup J vk ; BR (x0 ) ≤ lim sup k→∞

k→∞

BR (x0 )

BR (x0 )

f (∇umk ) dx ,

and (22) is established. From the definition of vk it also follows that lim |vk − u∗ | dx = 0 . k→∞

(24)

BR (x0 )

Note that by construction we clearly may assume that vk |BR/2 (x0 ) = umi |BR/2 (x0 ) for any k ∈ N and for some subsequence of {um }. Next, an analogous construction is needed in the exterior domain: choose some radius ρ > R, Bρ (x0 ) Ω, and a sequence u ˆm ∈ C ∞ (Bρ (x0 ) − BR (x0 ); RN ) satisfying u ˆm|∂(Bρ (x0 )−BR (x0 )) = u∗|∂(Bρ (x0 )−BR (x0 )) , and Bρ (x0 )−BR (x0 )

u ˆm → u∗ in L1 (Bρ (x0 ) − BR (x0 ); RN ) |∇ˆ um | dx → |∇u∗ | as m → ∞ . Bρ (x0 )−BR (x0 )

For small enough ε > 0 and for {vm }m∈N given in (22) we then let ⎧ ⎪ ⎪ ⎪ vm on BR (x0 ) ⎪ ⎪ ⎪ ⎨ ε := u wm ˆm )(|x| − R) on Tε − BR (x0 ) . ˆm + ε−1 (um − u ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ on the rest of Ω um

198


ε Again wm is seen to be of class W11 (Ω; RN ) and we have ε ; Ω] J[wm

ε = J vm ; BR (x0 ) + f (∇wm ) dx Tε −BR (x0 ) + f (∇um ) dx Ω∩[|x−x0 |>R+ε] = J vm ; BR (x0 ) + f (∇um ) dx Ω−BR (x0 ) ε + f (∇wm ) dx − f (∇um ) dx . Tε −BR (x0 )

(25)

Tε −BR (x0 )

As before we deduce from (20) the existence of sequences εk → 0 and mk → ∞ such that 2 2 and |∇um | dx ≤ |∇ˆ um | dx ≤ k k Tε −BR (x0 ) Tε −BR (x0 ) k

k

for all m ≥ mk , for all k ∈ N and we get 1 6 6 εk |∇wm | dx ≤ + |um − u ˆm | dx ≤ + εk , k εk Tε −BR (x0 ) k Tε −BR (x0 ) k

k

again for all k ∈ N and for all m ≥ mk . Thus decomposition (25) gives together with (22) for mk chosen sufficiently large εk ; Ω] J[wm

≤ J vm ; BR (x0 ) +

f (∇um ) dx + βk Ω−BR (x0 )

≤ lim inf J um ; BR (x0 ) + m→∞

f (∇um ) dx + 2βk Ω−BR (x0 )

for any k ∈ N, for all m ≥ mk and with a sequence βk → 0 as k → ∞. Again enlarging mk if necessary we may assume that εk J[wm ; Ω] ≤ lim inf J u ; B (x ) + lim inf J u ; Ω − B (x ) + 3βk m R 0 m R 0 k m→∞

m→∞

≤ lim inf J[um ; Ω] + 3βk . m→∞

εk Setting wk = wm the lemma is proved observing that as in (24) the definition k of wk also implies L1 convergence on the whole domain Ω. Moreover, it is clear that we may assume v).

C Brief comments on steady states of generalized Newtonian fluids

As one application of the above discussed methods let us give some short comments on steady states of generalized Newtonian fluids. Here, a non-uniform ellipticity condition fits into the main line of this monograph. On the other hand, solenoidal (divergence-free) vector-fields are considered and ∇u has to be replaced by its symmetric part ε(u). Besides of several technical difficulties (which we do not want to discuss) this, in particular, gives rise to a challenging open problem: do we expect that a structure condition f (ε) = g(|ε|2 ) in general is sufficient to imply full regularity of solutions in the above mentioned setting? Without going into the proofs let us give a short introduction to the results of [BF10] and [ABF] – the main ideas are closely related to the arguments of Section 3. The stationary flow of an incompressible generalized Newtonian fluid is considered in a bounded Lipschitz domain Ω ⊂ Rn , n = 2 or n = 3. To be precise, we are looking for a velocity field u: Ω → Rn solving the following system of nonlinear partial differential equations ⎫ k ∂u ⎪ + ∇π = g in Ω , −div T (ε(u)) + u ⎬ ∂xk (1) ⎪ ⎭ div u = 0 in Ω , u = 0 on ∂Ω . Here, π is the a priori unknown pressure function and g: Ω → Rn represents a system of volume forces. The tensor T is assumed to be the gradient of some (convex) potential f : S → [0, ∞) which is of class C 2 on the space S of all symmetric matrices. Again we adopt the convention of summation over repeated indices running from 1 to n, moreover, for functions v: Ω → Rn we let 1 j ∂i v + ∂j v i (x) ∈ S . ε(v)(x) = 2 2 In the case f (ε) = |ε| , the system (1) reduces to the Dirichlet (no-slip) boundary value problem for the stationary Navier-Stokes system, for an overview


200


on existence and regularity results we refer to the classical monograph [La] of Ladyzhenskaya or, more recently, to the monographs [Ga] of Galdi, where also the history of the problem is outlined in great detail. So-called power-law models are investigated for example in [KMS]: for some exponent 1 < p < ∞, f is assumed to satisfy λ(1 + |ε|2 )

p−2 2

|σ|2 ≤ D2 f (ε)(σ, σ) ≤ Λ(1 + |ε|2 )

p−2 2

|σ|2

(2)

for all ε, σ ∈ S and with positive constants λ, Λ. As above, (2) implies that f is of p-growth, moreover, the first inequality in (2) implies strict convexity of f . Then, if f (ε) = F (|ε|2 ) (which is reasonable from the physical point of view) Kaplick´ y, M´ alek and Star´ a discuss the two-dimensional case with the following results: if p > 3/2, then the problem (1) admits a solution which is of class C 1,α up to the boundary, whereas for p > 6/5 the problem (1) has a solution being C 1,α -regular in the interior of Ω. Here of course the volume force term g is sufficiently smooth. Suppose for the moment that the flow is also slow. Then in (1) the convective term (∇u)u = uk ∂k u can be neglected, and (1) reduces to a generalized version of the classical Stokes problem. In the monograph [FuS2], a variational approach towards (1) for various classes of dissipative potentials f is described leading to existence and also (partial) regularity results in the absence of the convective term. Very recently these investigations were extended in [BF10] to the case of non-uniformly elliptic potentials which means that (2) is replaced by the condition λ(1 + |ε|2 )

p−2 2

|σ|2 ≤ D2 f (ε)(σ, σ) ≤ Λ(1 + |ε|2 )

q−2 2

|σ|2

(3)

with exponents 1 < p ≤ q < ∞, q ≥ 2 and for all ε, σ ∈ S. From (3) it easily follows that f is of upper growth rate q, a lower bound for f (ε) can be given in terms of |ε|p (recall Remark 3.5). Examples of potentials f satisfying (3) are given in [BF10] following the lines of Section 3. Moreover, it is shown in this paper that weak local solutions of (1) (with (∇u)u = 0!) under condition (3) are C 1,α in the interior of Ω if n = 2, q = 2, and partially C 1,α if n = 3, provided that we impose the bound q < p(1 + 2/n). The objective of [ABF] is to study the anisotropic (with respect to the ellipticity condition) situation (3) for a non-vanishing convective term (∇u)u. To be more precise let us assume that g ∈ L∞ (Ω; Rn ) .

(4)

Remark C.1. For the sake of technical simplicity we just assume that the volume forces are bounded functions. Of course our results are also valid under the weaker assumption g ∈ Lt(p) (Ω; Rn ), whenever t(p) is chosen sufficiently large.


201

In order to get a weak form of (1) we multiply the first line of (1) with ϕ ∈ C0∞ (Ω; Rn ), div ϕ = 0, and obtain after an integration by parts (using div ϕ = 0) Df (ε(u)) : ε(ϕ) dx − u ⊗ u : ε(ϕ) dx = g · ϕ dx , (5) Ω

Ω

Ω

where u ⊗ v := (u v ). Thus, we have to solve the equation (5) together with div u = 0 in Ω, u = 0 on ∂Ω. A priori it is not clear to which space a weak solution should belong, therefore we give an existence proof by using an approximation procedure. To this purpose we again let (0 < δ < 1) i k

q

fδ (ε) := δ(1 + |ε|2 ) 2 + f (ε) ,

ε∈S,

and consider the problem

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

◦

uδ ∈Wq1 (Ω; Rn ),

to find div uδ = 0, such that DFδ (ε(uδ )) : ε(ϕ) dx − uδ ⊗ uδ : ε(ϕ) dx = g · ϕ dx ⎪ ⎪ Ω Ω Ω ⎪ ⎪ ⎪ ⎪ ⎭ ∞ n for all ϕ ∈ C0 (Ω; R ), div ϕ = 0.

(5δ )

Then we have Theorem C.2. Let f satisfy (3) with 1 < p < 2 ≤ q < ∞, and consider the ◦

data given in (4). Then (5δ ) has at least one weak solution uδ ∈Wq1 (Ω; Rn ). Moreover, we have sup uδ Wp1 (Ω;Rn ) < ∞ . 0 6/5 implies the solution to be of class L2 . It is not evident, how to apply this method to the non-standard models under consideration (see, for instance, formula (42) of [FrM]). Theorem C.4 just relies on the a priori estimates of Theorem C.3. ii) Note that, in contrast to Section 3, a global regularization is done. This is discussed in detail in [BF11]. As to the regularity properties of the particular solution u¯ from above, the considerations of [ABF] are restricted to the case q = 2. Theorem C.6. Under the assumptions and with the notation of Theorem C.4 we consider the case n = 3, q = 2. Then u ¯ is partially of class C 1,α , i.e. there ¯ ∈ is an open set Ω0 of full Lebesgue measure, |Ω − Ω0 | = 0, such that u 1,α 3 C (Ω0 ; R ). Remark C.7. As in [BF10] it is possible to give a variant of Theorem C.6 for the case q > 2 together with q < 5p/3. Theorem C.8. Under the assumptions and with the notation of Theorem C.4 we consider the case n = 2, q = 2. Then u ¯ has locally H¨ older continuous first derivatives, i.e. u ¯ ∈ C 1,α (Ω; R2 ).


203

Remark C.9. It would also be desirable to give global variants of the above results, for example to prove higher integrability of ∇¯ u up to the boundary. Then, under suitable smallness conditions, some results on unique solvability extend to our non-uniformly elliptic problems. The idea to establish a theorem of this kind is standard and, for instance, presented in [La], p. 118. The main difficulty in the case of non-uniform ellipticity is to handle potentials with lower growth rate p < 2.

D Notation and conventions

The set Ω ⊂ Rn , n ≥ 2, always denotes a bounded Lipschitz domain. For N ≥ 1, the reader is assumed to be familiar with the classical • H¨ older spaces C k,α Ω; RN , • Lebesgue spaces Lp Ω; RN as well as with the notion of the ◦ • Sobolev spaces Wpm Ω; RN , Wpm Ω; RN . Here we follow the definitions and notation as introduced in [GT], in particular local variants are denoted by • Lploc Ω; RN , etc. Moreover, the target space is not indicated whenever N = 1. The basics on •

◦

Orlicz-Sobolev spaces WF1 (Ω; RN ), WF1(Ω; RN )

are shortly outlined in Section 3.1, for more details we refer to [Ad]. Now let w ∈ L1 (Ω; RN ). Then w is called a • function of bounded variation in Ω, w ∈ BV Ω; RN , if the distributional derivative is representable by a finite Radon measure in Ω, i.e. for some RnN -valued measure (∂α wi )1≤i≤N 1≤α≤n we have

w div ϕ dx = − ϕiα ∇α wi Ω Ω =− ϕ : ∇w for any ϕ ∈ C01 Ω; RnN . i

w div ϕ dx = Ω

i

Ω

1n Here and in the following we have the conventions: div ϕ = ( α=1 ∂α ϕiα ∈ RN . Summation is always assumed with respect to repeated indices – for Latin indices the sum is taken over i = 1, . . . , N , for Greek indices this is done with


206

D Notation and conventions

repect to α = 1, . . . , n – and the scalar product in RN is not highlighted, whereas we take the symbol “:” for the standard scalar product in RnN . Moreover, with a slight abuse of notation, derivatives usually are denoted by “∇” – the precise meaning will always be evident by the context. The total variation of w is given by |∇w| = sup w div ϕ dx : ϕ ∈ C01 Ω; RnN , |ϕ(x)| ≤ 1 on Ω , Ω

Ω

and a L1 -function is seen to be of class BV (Ω; RN ) if and only if the total variation is finite. For these definitions and a variety of details we refer to [Giu2] and [AFP]. To consider functionals defined on measures (see [AFP], Section 2.6), let f : nN → [0, ∞) be strictly convex, continuous, of linear growth with f (0) = 0 R (of course, this assumption is too strong but always satisfied in our studies). The recession function f∞ : RnN → R is given by f∞ (Z) := lim sup t↑∞

f (t Z) f (t Z) = lim t↑∞ t t

for any Z ∈ RnN .

Obviously, f∞ is positively homogeneous of degree one. Now, any RnN -valued measure μ in Ω will always be decomposed with respect to the Lebesgue measure. Then μa denotes the absolutely continuous part, whereas μs denotes the singular part and μs /|μs | is the symbol for the Radon-Nikodym derivative. With this notation we may define (see [AFP], formula (2.26)) s a μ (x) d|μs |(x) . f μ (x) dx + f∞ (1) G(μ) := s |μ | Ω Ω Finally, if f ∗ denotes the conjugate of f , f ∗ (Q) := sup Z : Q − f (Z)

for any Q ∈ RnN ,

Z∈RnN

then we may let for any Borel set B ⊂ Ω FB (μ) :=

1B κ μ −

sup

κ∈C0∞ (Ω;RnN ), f ∗ ◦κ∈L1 (Ω)

Ω

f ∗ (κ) dx

.

(2)

Ω

Note that by Proposition 1.2, [DT], both viewpoints (1) and (2) coincide. It remains to mention three conventions which are not restated each time: • • •

if we do not give further comments, then any ball under consideration is assumed to be open and compactly contained in Ω; if necessary (and possible), we usually pass to subsequences without relabeling; positive constants are also not relabeled and usually just denoted by c, not necessarily being the same in any two occurrencies. Moreover, only relevant dependences are highlighted.

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Index

a priori gradient bound, 50, 101, 118, 129, 156, 170 absolutely continuous part of μ, 206 of ∇u∗ , 25 balancing condition, 4, 51, 98 blow-up, 9, 72, 144 bounded solution, 4, 10, 11, 104, 141, 170 Caccioppoli-type inequality, 54, 70, 146, 152, 166 limit-version, 70 Campanato estimate, 76 conjugate function, 8, 14, 177, 206 convective term, 200 convex hull property, 104, 130, 143

essentially smooth, 177 strictly convex, 177 Euler equation, 17, 30, 52, 108 excess function, 73 existence of solutions dual, 6, 17 energy class, 2 for (P ), 105, 123 generalized ˆ J-minimizer, 5, 175 ˜ J-minimizer, 176 u ∈ M, 5, 25, 100, 163, 173 obstacle problems, 150 Orlicz-Sobolev, 42

de la Vallèe Poussin criterion, 44 decay estimate, 72 degenerate set; u∗ , σ, 39 DeGiorgi’s technique, 9, 10, 65, 118, 156 difference quotient technique, 19, 55, 69 Dirichlet-growth theorem, 95 dissipative potential, 200 dual limit, 168 duality relation, 8, 15, 129 for degenerate problems, 33

Fatou’s lemma, 44, 55, 169 Frehse-Seregin Lemma, 95 Frehse-Seregin lemma, 10 function of bounded variation, 26, 205 BVu0 (Ω), 173 decomposition, 138 density results, 185 functional defined on measures, 206 dual, 16 relaxed ˆ 5, 26, 174 J, ˜ 176 J,

Egoroff’s theorem, 22, 34 ellipticity limit exponent, 10, 111, 123 μ, 4, 10, 48, 97

generalized minimizer u ∈ M, 5, 25, 100, 163, 173 generalized Newtonian fluid, 199 generalized Young’s inequality, 31

216

Index

growth condition anisotropic, 3, 41, 89, 91, 141 anisotropic linear/superlinear, 12, 161 first extension of the logarithm, 4 involving N-functions, 4, 90 linear, 5, 13, 97, 174 nearly linear, 4, 41 power, 2 (s,μ, q), 9, 44 H¨ older space, 205 higher integrability, 4, 54, 93, 111, 116, 123, 141, 144, 147, 154, 166 Hilbert-Haar theory, 138 hole filling technique, 60 identification theorem, 176 inf-sup relation, 19 interpolation inequality, 59 iteration lemma, 87 Jensen’s inequality, 187, 193 John-Nirenberg estimates, 100 Lp -theory, 53 Lagrangian, 5, 15, 164, 176 Lebesgue point, 34 Lebesgue space, 205 Lebesgue-Besicovitch Differentiation Theorem, 35 limit equation, 74 linearization, 151 local comparison function, 12, 26, 131, 194 local minimizer, 51, 144 lower semicontinuity, 27, 174 Lusin’s theorem, 35 Luxemburg norm, 42

partial regularity maximizer, 8, 29, 39 minimizer, 2, 8, 9, 26, 69, 144 perfect plasticity, 6, 176 plastic material with logarithmic hardening, 4, 6 plastic/elastic behavior, 161 Poincaré inequality, 44 Prandtl-Eyring fluid, 4 proper convex function, 177 closed, 177 Radon-Nikodym derivative, 206 recession function, 27, 105, 206 regular set Ωu∗ , 25 regularization, 16, 52, 69, 92, 101, 108, 146, 151, 166 representation formula ˆ 27, 174 for J, ˜ 180 for J, for J, 15, 164 Reshetnyak’s lower semicontinuity theorem, 174

N-function, 4, 42, 142 Δ2 -condition, 4, 42, 142

(s,μ, q)-condition, 9, 51 singular part of μ, 206 of ∇u∗ , 25 Sobolev space, 205 Sobolev’s inequality on minimal hypersurfaces, 99 Sobolev-Poincaré type inequality, 81 Sobolev-Poincarè inequality, 124 solenoidal, 199 starting inequality via Euler equation, 111, 116, 124, 126 integration by parts, 147, 154 Sobolev’s inequality, 58 Stokes problem, 200 stress tensor, 6 local, 12, 172 stress-strain relation, 172 structure condition geometric, 6, 10, 99, 106 N > 1, 2, 50, 62, 100, 109, 128

obstacle problem, 51, 149 Orlicz space, 42 Orlicz-Sobolev space, 42, 205

Theorem on Domain Invariance, 21 Theorem on Dominated Convergence, 57

maximum principle, 104, 130, 143 minimax inequality, 29, 170, 181 Moser iteration, 156

Index total variation, 206 two-dimensional problems, 3, 9, 10, 91, 122 uniqueness of solutions dual, 6, 8, 20 energy class, 2 for (P ), 10, 105, 123, 183 generalized, 6, 10, 100, 106, 123, 164, 182 obstacle problems, 150 Orlicz-Sobolev, 42 variational integrand asymptotically regular, 111, 146 linear growth, 13, 97, 174 minimal surface, 6, 10, 99, 106, 174 quasiconvex, 1

колхоз 1:12 am, 7/23/05

217

(s,μ, q)-growth, 44 strictly convex classification, 2 GEN. ASS, 1 with a wide range of anisotropy, 141, 161 x-dependent, 11, 113, 132 variational problem (P), 1, 43, 97, 142, 163 (P ), 10, 99, 122, 181 degenerate, 8, 32, 150 dual (P ∗ ), 5, 14, 101 obstacle, 51, 149 relaxed ˆ 5, 175 (P), ˜ 176 (P), Vitali’s convergence theorem, 55

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