Applied Mathematical Sciences

By: Bernard Dacorogna

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Hardcover | 13 December 2007 | Edition Number 2

At a Glance

Hardcover
636 Pages

Edition Type
Revised

Dimensions(cm)
23.5 x 15.88 x 3.18

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This second edition is the successor to "Direct methods in the calculus of variations" which was published in the Applied Mathematical Sciences series and is currently out of print. Although the core and the structure of the present book is similar to the first edition, it is much more than a revised version. Fifteen years have passed since the publication of the "Direct methods in the calculus of variations" book and since the subject is a very active one, almost half of the book presently consists of new material. The perspective has also slightly changed, indeed, a new subject, "quasiconvex analysis" has now been developed. The present edition, which is essentially a reference book on the subject of quasiconvex analysis can be used, as was the earlier book, for an advanced course on the calculus of variations.

Industry Reviews

From the reviews of the second edition:

"This new edition is very much expanded, up to the point that it can be considered an almost entirely new book. ... I was a PhD student in the early 90's, and I have been studying and using it continuously since then. It has accompanied me and many others as a valuable source for results and as a standard reference on the subject. I am sure that this new edition will be serving in the same role as well." (Pietro Celada, Mathematical Reviews, Issue 2008 m)

"The present monograph has been ... a 'revised and augmented edition to Direct Methods in the Calculus of Variations'. ... the author maintains a fresh and lucid style, resulting in a concise, very well readable presentation. Surely this book will define a long-lasting standard in its area. ... The exhaustive bibliography comprises 621 references and covers the relevant publications in the area ... ." (Marcus Wagner, Zentralblatt MATH, Vol. 1140, 2008)

"This is a substantially extended new edition of the author's introduction to direct methods in the calculus of variations. ... The author has taken great care to include all the main developments in the area since the first edition (the list of references comprises 621 items). The book is carefully written and provides a very readable introduction to the field." (M. Kunzinger, Monatshefte fuer Mathematik, Vol. 160 (4), July, 2010)

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Calculus Differential Calculus & Equations Optimisation Linear Programming Calculus of Variations Functional Analysis & Transforms Applied Mathematics

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Published: 29th November 2010

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

At a Glance

Hardcover

Industry Reviews

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