Preface ........................................................ xi
1. Introduction ................................................ 1
1.1. The direct methods of the calculus of variations ...... 1
1.2. Convex analysis and the scalar case ................... 3
1.2.1. Convex analysis ................................ 4
1.2.2. Lower seniicontinuity and existence results .... 5
1.2.3. The one dimensional case ....................... 7
1.3. Quasiconvex analysis and the vectorial case ........... 9
1.3.1. Quasiconvex functions .......................... 9
1.3.2. Quasiconvex envelopes ......................... 12
1.3.3. Quasiconvex sets .............................. 13
1.3.4. Lower seniicontinuity and existence
theorems ...................................... 15
1.4. Relaxation and non-convex problems ................... 17
1.4.1. Relaxation theorems ........................... 18
1.4.2. Some existence theorems for differential
inclusions .................................... 19
1.4.3. Some existence results for non-quasiconvex
integrands .................................... 20
1.5. Miscellaneous ........................................ 23
1.5.1. Hölder and Sobolev spaces ..................... 23
1.5.2. Singular values ............................... 23
1.5.3. Some underdetermincd partial differential
equations ..................................... 24
1.5.4. Extension of Lipschitz maps ................... 25
I. CONVEX ANALYSIS AND THE SCALAR CASE ........................ 29
2. Convex sets and convex functions ........................... 31
2.1. Introduction ......................................... 31
2.2. Convex sets .......................................... 32
2.2.1. Basic definitions and properties .............. 32
2.2.2. Separation theorems ........................... 34
2.2.3. Convex hull and Carathcodory theorem .......... 38
2.2.4. Extreme points and Minkowski theorem .......... 42
2.3. Convex functions ..................................... 44
2.3.1. Basic definitions and properties .............. 44
2.3.2. Continuity of convex functions ................ 46
2.3.3. Convex envelope ............................... 52
2.3.4. Lower semicontinuous envelope ................. 56
2.3.5. Legendre transform and duality ................ 57
2.3.6. Subgradients and differentiability
of convex functions ........................... 61
2.3.7. Gauges and their polars ....................... 68
2.3.8. Choquet function .............................. 70
3. Lower semicontinuity and existence theorems ................ 73
3.1. Introduction ......................................... 73
3.2. Weak lower semicontinuity ............................ 74
3.2.1. Preliminaries ................................. 74
3.2.2. Some approximation lemmas ..................... 77
3.2.3. Necessary condition: the case without lower
order terms ................................... 82
3.2.4. Necessary condition: the general case.......... 84
3.2.5. Sufficient condition: a particular case ....... 94
3.2.6. Sufficient condition: the general case ........ 96
3.3. Weak continuity and invariant integrals ............. 101
3.3.1. Weak continuity .............................. 101
3.3.2. Invariant integrals .......................... 103
3.4. Existence theorems and Euler-Lagrange equations ..... 105
3.4.1. Existence theorems ........................... 105
3.4.2. Euler-Lagrange equations ..................... 108
3.4.3. Some regularity results ...................... 116
4. The one dimensional case .................................. 119
4.1. Introduction ........................................ 119
4.2. An existence theorem ................................ 120
4.3. The Euler-Lagrange equation ......................... 125
4.3.1. The classical and the weak forms ............. 125
4.3.2. Second form of the Euler-Lagrange equation ... 129
4.4. Some inequalities ................................... 132
4.4.1. Pohicare-Wirtinger inequality ................ 132
4.4.2. Wirtinger inequality ......................... 132
4.5. Hamiltonian formulation ............................. 137
4.6. Regularity .......................................... 143
4.7. Lavrentiev phenomenon ............................... 148
II. QUASICONVEX ANALYSIS AND THE VECTORIAL CASE ............... 153
5. Polyconvex, quasiconvex and rank one convex functions ..... 155
5.1. Introduction ........................................ 155
5.2. Definitions and main properties ..................... 156
5.2.1. Definitions and notations .................... 156
5.2.2. Main properties .............................. 158
5.2.3. Further properties of polyconvex functions ... 163
5.2.4. Further properties of quasiconvex
functions .................................... 171
5.2.5. Further properties of rank one convex
functions .................................... 174
5.3. Examples ............................................ 178
5.3.1. Quasiaffine functions ........................ 179
5.3.2. Quadratic case ............................... 191
5.3.3. Convexity of SO(n) x SO(n) and O(N) x O(n)
invariant functions .......................... 197
5.3.4. Polyconvexity and rank one convexity of
SO(n) x SO(n) and O(N) x O(n) invariant
functions .................................... 202
5.3.5. Functions depending on a quasiaffine
function ..................................... 212
5.3.6. The area type case ........................... 215
5.3.7. The example of Sverak ........................ 219
5.3.8. The example of Alibert-Dacorogna-
Marcellini ................................... 221
5.3.9. Quasiconvex functions with subquadratic
growth ....................................... 237
5.3.10.The case of homogeneous functions of
degree one ................................... 239
5.3.11.Sonic-more examples .......................... 245
5.4. Appendix: some basic properties of determinants ..... 249
6. Polyconvex, quasiconvex and rank one convex envelopes ..... 265
6.1. Introduction ........................................ 265
6.2. The polyconvex envelope ............................. 266
6.2.1. Duality for polyconvex functions ............ 266
6.2.2. Another representation formula .............. 269
6.3. The quasiconvex envelope ............................ 271
6.4. The rank one convex envelope ........................ 277
6.5. Some more properties of the envelopes ............... 280
6.5.1. Envelopes and sums of functions .............. 280
6.5.2. Envelopes and invariances .................... 282
6.6. Examples ............................................ 285
6.6.1. Duality for SO(n) x SO(n) and O(N) x O(n)
invariant functions .......................... 285
6.6.2. The case of singular values .................. 291
6.6.3. Functions depending on a quasiaffine
function ..................................... 296
6.6.4. The area type case ........................... 298
6.6.5. The Kohn-Strang example ...................... 300
6.6.6. The Saint Venant-Kirchhoff energy function ... 305
6.6.7. The case of a norm ........................... 309
7. Polyconvex, quasiconvex and rank one convex sets .......... 313
7.1. Introduction ........................................ 313
7.2. Polyconvex, quasiconvex and rank one convex sets .... 315
7.2.1. Definitions and main properties .............. 315
7.2.2. Separation theorems for polyconvex sets ...... 321
7.2.3. Appendix: functions with finitely many
gradients .................................... 322
7.3. The different types of convex hulls ................. 323
7.3.1. The different convex hulls ................... 323
7.3.2. The different convex finite hulls ............ 331
7.3.3. Extreme points and Minkowski type theorem
for polyconvex, quasiconvex and rank one
convex sets .................................. 335
7.3.4. Gauges for polyconvex sets ................... 342
7.3.5. Choquet functions for polyconvex and rank
one convex sets .............................. 344
7.4. Examples ............................................ 347
7.4.1. The case of singular values .................. 348
7.4.2. The case of potential wells .................. 355
7.4.3. The case of a quasiaffine function ........... 362
7.4.4. A problem of optimal design .................. 364
8. Lower semi continuity and existence theorems in the
vectorial case ............................................ 367
8.1. Introduction ........................................ 367
8.2. Weak lower scmicontinuity ........................... 368
8.2.1. Necessary condition .......................... 368
8.2.2. Lower semicontinuity for quasiconvex
functions without lower order terms .......... 369
8.2.3. Lower semicontinuity for general
quasiconvex functions for p = ∞ ............. 377
8.2.4. Lower semicontinuity for general
quasiconvex functions for 1 < p < ∞ .......... 381
8.2.5. Lower semicontinuity for polyconvex
functions .................................... 391
8.3. Weak Continuity ..................................... 393
8.3.1. Necessary condition .......................... 393
8.3.2. Sufficient condition ......................... 394
8.4. Existence theorems .................................. 403
8.4.1. Existence theorem for quasiconvex
functions .................................... 403
8.4.2. Existence theorem for polyconvex functions ... 404
8.5. Appendix: some properties of Jacobians .............. 407
III.RELAXATION AND NON-CONVEX PROBLEMS ........................ 413
9. Relaxation theorems ....................................... 415
9.1. Introduction ........................................ 415
9.2. Relaxation Theorems ................................. 416
9.2.1. The case without lower order terms ........... 416
9.2.2. The general case ............................. 424
10. Implicit partial differential equations ................... 439
10.1. Introduction ........................................ 439
10.2. Existence theorems .................................. 440
10.2.1. An abstract theorem ......................... 440
10.2.2. A sufficient condition for the relaxation
property .................................... 444
10.2.3. Appendix: Baire one functions ............... 449
10.3. Examples ............................................ 451
10.3.1. The scalar case ............................. 451
10.3.2. The case of singular values ................. 459
10.3.3. The case of potential wells ................. 461
10.3.4. The case of a quasiaffine function .......... 462
10.3.5. A problem of optimal design ................. 463
11. Existence of minima for non-quasiconvex integrands ........ 465
11.1. Introduction ........................................ 465
11.2. Sufficient conditions ............................... 467
11.3. Necessary conditions ................................ 472
11.4. The scalar case ..................................... 483
11.4.1. The case of single integrals ................ 483
11.4.2. The case of multiple integrals .............. 485
11.5. The vectorial case .................................. 487
11.5.1. The case of singular values ................. 488
11.5.2. The case of quasiaffine functions ........... 490
11.5.3. The Saint Venant-Kirchhoff energy ........... 492
11.5.4. A problem of optimal design ................. 493
11.5.5. The area type case .......................... 494
11.5.6. The case of potential wells ................. 498
IV. MISCELLANEOUS ............................................. 501
12. Function spaces ........................................... 503
12.1. Introduction ........................................ 503
12.2. Main notation ....................................... 503
12.3. Some properties of Holder spaces .................... 506
12.4. Some properties of Sobolev spaces ................... 509
12.4.1. Definitions and notations ................... 510
12.4.2. Imbeddings and compact imbeddings ........... 510
12.4.3. Approximation by smooth and piecewise
affine functions ............................ 512
13. Singular values ........................................... 515
13.1. Introduction ........................................ 515
13.2. Definition and basic properties ..................... 515
13.3. Signed singular values and von Neumann type
inequalities ........................................ 519
14. Some underdetermined partial differential equations ....... 529
14.1. Introduction ........................................ 529
14.2. The equations div u = f and curl u = f .............. 529
14.2.1. A preliminary lemma ......................... 529
14.2.2. The case div u = f .......................... 531
14.2.3. The case curl u = f ......................... 533
14.3. The equation det Ñ u = f ............................ 535
14.3.1. The main theorem and some corollaries ....... 535
14.3.2. A deformation argument ...................... 539
14.3.3. A proof under a smallness assumption ........ 541
14.3.4. Two proofs of the main theorem .............. 543
15. Extension of Lipschitz functions on Banach spaces ......... 549
15.1. Introduction ........................................ 549
15.2. Preliminaries and notation .......................... 549
15.3. Norms induced by an inner product ................... 551
15.4. Extension from a general subset of E to E ........... 558
15.5. Extension from a convex subset of E to E ............ 565
Bibliography .................................................. 569
Notation ...................................................... 611
Index ......................................................... 615
|
