Direct Methods in the Calculus of Variations Book

Preface     xi
Introduction     1
The direct methods of the calculus of variations     1
Convex analysis and the scalar case     3
Convex analysis     4
Lower semicontinuity and existence results     5
The one dimensional case     7
Quasiconvex analysis and the vectorial case     9
Quasiconvex functions     9
Quasiconvex envelopes     12
Quasiconvex sets     13
Lower semicontinuity and existence theorems     15
Relaxation and non-convex problems     17
Relaxation theorems     18
Some existence theorems for differential inclusions     19
Some existence results for non-quasiconvex integrands     20
Miscellaneous     23
Holder and Sobolev spaces     23
Singular values     23
Some underdetermined partial differential equations     24
Extension of Lipschitz maps     25
Convex analysis and the scalar case     29
Convex sets and convex functions     31
Introduction     31
Convex sets     32
Basic definitions and properties     32
Separation theorems     34
Convex hull and Caratheodory theorem     38
Extreme points and Minkowski theorem     42
Convex functions     44
Basic definitions and properties     44
Continuity of convex functions     46
Convex envelope     52
Lower semicontinuous envelope     56
Legendre transform and duality     57
Subgradients and differentiability of convex functions     61
Gauges and their polars     68
Choquet function     70
Lower semicontinuity and existence theorems     73
Introduction     73
Weak lower semicontinuity     74
Preliminaries     74
Some approximation lemmas     77
Necessary condition: the case without lower order terms     82
Necessary condition: the general case     84
Sufficient condition: a particular case     94
Sufficient condition: the general case     96
Weak continuity and invariant integrals     101
Weak continuity     101
Invariant integrals     103
Existence theorems and Euler-Lagrange equations     105
Existence theorems     105
Euler-Lagrange equations     108
Some regularity results     116
The one dimensional case     119
Introduction     119
An existence theorem     120
The Euler-Lagrange equation     125
The classical and the weak forms     125
Second form of the Euler-Lagrange equation     129
Some inequalities     132
Poincare-Wirtinger inequality     132
Wirtinger inequality     132
Hamiltonian formulation     137
Regularity     143
Lavrentiev phenomenon     148
Quasiconvex analysis and the vectorial case     153
Polyconvex, quasiconvex and rank one convex functions     155
Introduction     155
Definitions and main properties     156
Definitions and notations     156
Main properties     158
Further properties of polyconvex functions     163
Further properties of quasiconvex functions     171
Further properties of rank one convex functions     174
Examples     178
Quasiaffine functions     179
Quadratic case     191
Convexity of SO (n) x SO (n) and O (N) x O (n) invariant functions     197
Polyconvexity and rank one convexity of SO (n) x SO (n) and O (N) x O (n) invariant functions     202
Functions depending on a quasiaffine function     212
The area type case     215
The example of Sverak     219
The example of Alibert-Dacorogna-Marcellini     221
Quasiconvex functions with subquadratic growth     237
The case of homogeneous functions of degree one     239
Some more examples     245
Appendix: some basic properties of determinants     249
Polyconvex, quasiconvex and rank one convex envelopes     265
Introduction     265
The polyconvex envelope     266
Duality for polyconvex functions     266
Another representation formula     269
The quasiconvex envelope     271
The rank one convex envelope     277
Some more properties of the envelopes     280
Envelopes and sums of functions     280
Envelopes and invariances     282
Examples     285
Duality for SO (n) x SO (n) and O (N) x O (n) invariant functions     285
The case of singular values     291
Functions depending on a quasiaffine function     296
The area type case     298
The Kohn-Strang example      300
The Saint Venant-Kirchhoff energy function     305
The case of a norm     309
Polyconvex, quasiconvex and rank one convex sets     313
Introduction     313
Polyconvex, quasiconvex and rank one convex sets     315
Definitions and main properties     315
Separation theorems for polyconvex sets     321
Appendix: functions with finitely many gradients     322
The different types of convex hulls     323
The different convex hulls     323
The different convex finite hulls     331
Extreme points and Minkowski type theorem for polyconvex, quasiconvex and rank one convex sets     335
Gauges for polyconvex sets     342
Choquet functions for polyconvex and rank one convex sets     344
Examples     347
The case of singular values     348
The case of potential wells     355
The case of a quasiaffine function     362
A problem of optimal design     364
Lower semi continuity and existence theorems in the vectorial case     367
Introduction     367
Weak lower semicontinuity     368
Necessary condition     368
Lower semicontinuity for quasiconvex functions without lower order terms     369
Lower semicontinuity for general quasiconvex functions for p = [infinity]     377
Lower semicontinuity for general quasiconvex functions for 1 [less than or equal] p< [infinity]     381
Lower semicontinuity for polyconvex functions     391
Weak Continuity     393
Necessary condition     393
Sufficient condition     394
Existence theorems     403
Existence theorem for quasiconvex functions     403
Existence theorem for polyconvex functions     404
Appendix: some properties of Jacobians     407
Relaxation and non-convex problems     413
Relaxation theorems     415
Introduction     415
Relaxation Theorems     416
The case without lower order terms     416
The general case     424
Implicit partial differential equations     439
Introduction     439
Existence theorems     440
An abstract theorem     440
A sufficient condition for the relaxation property     444
Appendix: Baire one functions     449
Examples     451
The scalar case     452
The case of singular values      459
The case of potential wells     402
The case of a quasiaffine function     462
A problem of optimal design     453
Existence of minima for non-quasiconvex integrands     465
Introduction     465
Sufficient conditions     457
Necessary conditions     472
The scalar case     433
The case of single integrals     483
The case of multiple integrals     485
The vectorial case     437
The case of singular values     488
The case of quasiaffine functions     490
The Saint Venant-Kirchhoff energy     492
A problem of optimal design     493
The area type case     494
The case of potential wells     498
Miscellaneous     501
Function spaces     503
Introduction     503
Main notation     503
Some properties of Holder spaces     506
Some properties of Sobolev spaces     509
Definitions and notations     510
Imbeddings and compact imbeddings     510
Approximation by smooth and piecewise affine functions     512
Singular values     515
Introduction      515
Definition and basic properties     515
Signed singular values and von Neumann type inequalities     519
Some underdetermined partial differential equations     529
Introduction     529
The equations div u = f and curl u = f     529
A preliminary lemma     529
The case div u = f     531
The case curl u = f     533
The equation det [nabla] u = f     535
The main theorem and some corollaries     535
A deformation argument     539
A proof under a smallness assumption     541
Two proofs of the main theorem     543
Extension of Lipschitz functions on Banach spaces     549
Introduction     549
Preliminaries and notation     549
Norms induced by an inner product     551
Extension from a general subset of E to E     558
Extension from a convex subset of E to E     565
Bibliography     569
Notation     611
Index     615