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Dacorogna B. — Direct Methods in the Calculus of Variations
Dacorogna B. — Direct Methods in the Calculus of Variations

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Название: Direct Methods in the Calculus of Variations

Автор: Dacorogna B.

Аннотация:

This monograph studies vectorial problems in the calculus of variations and quasiconvex analysis. It is a new edition of the earlier book published in 1989 and has been updated with some new material and examples added. This book will appeal to researchers and graduate students in mathematics and engineering.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: Second Edition

Год издания: 2007

Количество страниц: 619

Добавлена в каталог: 07.05.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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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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