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| Dacorogna B. — Direct Methods in the Calculus of Variations |
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| Предметный указатель |
Adjugate matrix 250 517
Affine hull, definition 32
Affine set 32
Affine set, definition 32
Alibert — Dacorogna — Marcellini example 179 221
Anticonformal part of a matrix, definition 224
Approximation property 445 446 448 460 462 463
Approximation property, definition 444
Baire category theorem 440 443 450
Baire one function 449 450
Baire one function, definition 449
Betti numbers 535
Biconjugate function, definition 57
Bidual function, definition 57
Bolza example 123 484
Brachistochrone 2 127
Canonical form 138
Caratheodory function, definition 75
Caratheodory theorem 38 39 41 52 54 56 158 163—167 265 288 318 323 480—482 484
Cauchy — Riemann equations 229
Cauchy — Schwarz inequality 36 68 567
Characteristic function, definition 75
Choquet function for polyconvex sets, definition 345
Choquet function for rank one convex sets, definition 346
Choquet function, definition 70
Conformal part of a matrix, definition 224
Conjugate function 62 342
Conjugate function, definition 57
Contraction map, definition 549
Controllable growth condition 111
Convergence in the sense of distributions, definition 393
Convex envelope 55 58 238 265 287 416 425 432 483 490
Convex envelope, definition 52
Convex function 207
Convex function, definition 44
Convex hull 55 58 165 317 327 338 364 439 440 452 551
Convex hull, definition 38
Convex integration 461
Convex set 32 33 35—37 42 44 45 64 68 70 318 320 335 337 357 442 550
Convex set, definition 32
Cycloid 128
Dacorogna formula 271
Dacorogna — Moser theorem 536
Directional derivative, definition 62
Dirichlet integral 2 6 106
Distance function 56 531
Distance function, definition 45
Divergence theorem 532
Domain of a function, definition 45
Dual function 69
Dual function, definition 57
DuBois — Reymond equation 129
Dunford — Pettis theorem 506
Egorov theorem 76
Ellipticity condition 115 117 155 159
Epigraph, definition 45
Equiintegrability 371 376 384 386 389 391
Equiintegrability, definition 506
Euler — Lagrange equation 3 111 117 119 126—131 137 138 141—144 146—148 155 159 178 191
Extension property for contractions 559 560 562
Extension property for contractions, , definition 550
Extreme point 42 43 70
Extreme point in the convex sense 337
Extreme point in the convex sense, definition 335
Extreme point in the polyconvex sense 337 338 344
Extreme point in the polyconvex sense, definition 335
Extreme point in the quasiconvex sense 340
Extreme point in the quasiconvex sense, definition 335
Extreme point in the rank one convex sense 338 340 346
Extreme point in the rank one convex sense, definition 336
Extreme point in the separately convex sense 337 340
Extreme point in the separately convex sense, definition 336
Extreme point, definition 42
Fatou lemma 95
Fermat principle 2
Field theory 128
Fourier series 134
Fourier transform 193
Fundamental lemma of the calculus of variations 115
Gateaux derivative 108
Gauge 68 342 552 555 558 567
Gauge in the polyconvex sense 343
Gauge in the polyconvex sense, definition 342
Gauge, definition 45 68
Generalized von Neumann inequality 520 521 523
Green formula 557
Hahn — Banach theorem 34 77
Hamiltonian 120 138 142 143
Hamiltonian system 142 143 146
Harmonic field 535
hessian 66
Hoelder continuous function 506 507 541
Hoelder inequality 107 111 113 135 374 376 399 400 422
Hoelder norm 59 69 559 561
Hoelder space 529
Hoelder space, definition 506
Hyperplane, definition 32
Indicator function 45 55 57 58 70 318 328 329 342 345
Indicator function, definition 44
Invariant integral 103 129
Isoperimetric inequality 120 132 137
Isotropic function, definition 200
Jensen inequality 46 124 127 150 161 418 419 458 481 538
Jordan-von Neumann theorem 554 556
Kirszbraun theorem 558 563
Kohn — Strang example 300
Kostant theorem 200
Krein — Milman theorem 42
Lagrangian 8 120 143
Laminate convex hull 331
Laplace equation 7 228
Lavrentiev phenomenon 120 148
Lebesgue dominated convergence theorem 451
Lebesgue monotone convergence theorem 99 101 379 383
Legendre transform 120 138
Legendre transform, definition 57
Legendre — Hadamard condition 159 162 220 225 242
Level set, definition 45
Lipschitz boundary, definition 510
Lipschitz continuous function 507
Loewner theorem 551
Lower semicontinuous envelope 58
Lower semicontinuous envelope, definition 56
Lower semicontinuous function 46
Lower semicontinuous function, definition 45
Lusin theorem 76
MacShane lemma 558 560 561
Mania example 148
| Matrix of cofactors 253
Mazur theorem 77 96
Minimal surface, non parametric form 2 106
Minimal surface, of revolution 128
Minimal surface, parametric form 2 13 419
Minimax theorem 550
Minkowski function, definition 68
Minkowski theorem 42 338 339
Minty theorem 550
Natural growth condition 111
Newton problem 2
Nordlander inequality 556 561
Normal integrand 94
Normal integrand, definition 74
Null Lagrangian 129 157
One sided directional derivative, definition 62
Orthogonal matrix, definition 515
Oscillation of a function, definition 449
Parallelogram rule 554 556 566
Parseval formula 134
Piecewise continuous function, definition 505
Plancherel formula 194
Poincare inequality 107 406
Poincare — Wirtinger inequality 120 123 132 133
Polar function 68 552 567
Polar function, definition 68
Polyaffine function 158 179
Polyaffine function, definition 157
Polyconvex biconjugate function, definition 267
Polyconvex conjugate function, definition 266
Polyconvex envelope 269 288 328
Polyconvex envelope, definition 265
Polyconvex finite hull, definition 331
Polyconvex function 158 159 161—164 169 171 174 179 182 192—194 196 203—210 213 215 222 226 228 237 239 246 248 266—271 289 291 301 318 344 351 391 403 404 474 478 479
Polyconvex function, definition 157
Polyconvex hull 328 332 349 440
Polyconvex hull, definition 323
Polyconvex set 317—321 325 329 333 335 342—344
Polyconvex set, definition 316
Potential wells 355 440 461 488 498
Projection map 35 565
Proper separating hyperplane, definition 35
Pyramid 452
Quasiaffine function 157 158 178—181 212 213 229 296 297 322 362 368 393 394 417 421 462 467—470 488 490—492 494 496
Quasiaffine function, definition 157
Quasiconvex envelope 238 415 416 432 466 493
Quasiconvex envelope, definition 265
Quasiconvex finite hull 440
Quasiconvex finite hull, definition 331
Quasiconvex function 155 157—159 161 162 169 171 173 178 191 192 194 212 213 215 219 222 231 237 239 246 271 272 274—276 291 318 367—373 377 378 381 382 384 390 391 394 403 404 415 424 441 442 465 472
Quasiconvex function, definition 156
Quasiconvex hull 325 332 339 349
Quasiconvex hull, definition 323
Quasiconvex set 319 326 329 333 335
Quasiconvex set, definition 316
Rademacher theorem 451 509
Rank one affine function 158 179 181—189
Rank one affine function, definition 157
Rank one convex envelope 295 328 329
Rank one convex envelope, definition 265
Rank one convex finite hull, definition 331
Rank one convex function 155 157 159—162 169 171 174—176 178 179 191 192 194 195 212—216 219 220 222 225 226 239 241 242 246—249 276—279 291 293 318 346
Rank one convex function, definition 156
Rank one convex hull 293 326 328 332 339 349 364 440 446
Rank one convex hull, definition 323
Rank one convex set 318 319 329 333 336 340 346 347 362 446
Rank one convex set, definition 316
Relative boundary of a convex set, definition 32
Relative interior of a convex set, definition 32
Relaxation property 441 443—445 467—471
Relaxation property, definition 441
Relaxation theorem 319 416 432 439 465 467 498
Relaxed problem 415 473 490 495
Relaxed problem, definition 416
Rellich theorem 98 388 512
Saint Venant — Kirchhoff energy function 305 488 492
Scorza — Dragoni theorem 76 99 385 427
Separately convex finite hull, definition 331
Separately convex function 47 48 51 158 160 162 239 318
Separately convex function, definition 47 157
Separately convex hull 327 332 339
Separately convex hull, definition 323
Separately convex set 318 329 333 336 337
Separately convex set, definition 316
Separating hyperplane, definition 35
Separation theorem 36 158 163—165 265 321 458
Set of permutation matrices, definition 197
Signed singular values 197 200 285
Signed singular values, definition 519
Simple function 75
Singular values 197 202 203 212 285 291 348 440 459 488 517 518 526
Singular values decomposition theorem 198 199 218 516
Singular values, definition 516
Sobolev exponent 511
Sobolev imbedding theorem 109 111 114 117 399 400 511
Sobolev space, definition 510
Special orthogonal matrix, definition 515
Stationary point 138
Strictly convex function 35 36 70 106 121 139 142 147 346 347 559
Strictly convex function in at least N directions 474 476 480
Strictly convex function in at least N directions, definition 474
Strictly convex function, definition 44
Strictly quasiconvex function 472—474 476 478 492 495 496 498
Strictly quasiconvex function, definition 472
Strictly separating hyperplane 35
Subdifferential 62 567
Subdifferential, definition 61
Subgradient 61 63 64
Subgradient, definition 61
Support function 58
Support function, definition 45
Sverak example 159 173 179 219
Tietze extension theorem 88 116
Tonelli partial regularity theorem 148
Variations of independent variables 130
Vitali covering theorem 453 457
von Neumann inequality 200 520 525 526
Weak continuity 368 392 393
Weak form of Euler — Lagrange equation 119 125 126
Weak lower semicontinuity 85 95 97 367 369 377 392
Weak lower semicontinuity, definition 74
Weak* lower semicontinuity 82—85 87 88 92 97 368 378
Weak* lower semicontinuity, definition 74
Weierstrass example 122
Weyl theorem 527
Wirtinger inequality 120 132 133 137
Young measure 418
Zorn Lemma 563
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