Dacorogna Direct Methods in the Calculus of Variations

2. Auflage 2007
ISBN: 978-0-387-55249-1
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 622 Seiten

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.

Autoren/Hrsg.

Dacorogna, Bernard

Inhaltsverzeichnis

1;Contents;6
2;Preface;12
3;Introduction;14
3.1;1.1 The direct methods of the calculus of variations;14
3.2;1.2 Convex analysis and the scalar case;16
3.3;1.3 Quasiconvex analysis and the vectorial case;22
3.4;1.4 Relaxation and non-convex problems;30
3.5;1.5 Miscellaneous;36
4;Convex analysis and the scalar case;41
4.1;Convex sets and convex functions;42
4.1.1;2.1 Introduction;42
4.1.2;2.2 Convex sets;43
4.1.3;2.3 Convex functions;55
4.2;Lower semicontinuity and existence theorems;83
4.2.1;3.1 Introduction;83
4.2.2;3.2 Weak lower semicontinuity;84
4.2.3;3.3 Weak continuity and invariant integrals;111
4.2.4;3.4 Existence theorems and Euler-Lagrange equations;115
4.3;The one dimensional case;128
4.3.1;4.1 Introduction;128
4.3.2;4.2 An existence theorem;129
4.3.3;4.3 The Euler-Lagrange equation;134
4.3.4;4.4 Some inequalities;141
4.3.5;4.5 Hamiltonian formulation;146
4.3.6;4.6 Regularity;152
4.3.7;4.7 Lavrentiev phenomenon;157
5;Quasiconvex analysis and the vectorial case;161
5.1;Polyconvex, quasiconvex and rank one convex functions;162
5.1.1;5.1 Introduction;162
5.1.2;5.2 Definitions and main properties;163
5.1.3;5.3 Examples;185
5.1.4;5.4 Appendix: some basic properties of determinants;256
5.2;Polyconvex, quasiconvex and rank one convex envelopes;271
5.2.1;6.1 Introduction;271
5.2.2;6.2 The polyconvex envelope;272
5.2.3;6.3 The quasiconvex envelope;277
5.2.4;6.4 The rank one convex envelope;283
5.2.5;6.5 Some more properties of the envelopes;286
5.2.6;6.6 Examples;291
5.3;Polyconvex, quasiconvex and rank one convex sets;319
5.3.1;7.1 Introduction;319
5.3.2;7.2 Polyconvex, quasiconvex and rank one convex sets;321
5.3.3;7.3 The different types of convex hulls;329
5.3.4;7.4 Examples;353
5.4;Lower semi continuity and existence theorems in the vectorial case;373
5.4.1;8.1 Introduction;373
5.4.2;8.2 Weak lower semicontinuity;374
5.4.3;8.3 Weak Continuity;399
5.4.4;8.4 Existence theorems;409
5.4.5;8.5 Appendix: some properties of Jacobians;413
6;Relaxation and non-convex problems;418
6.1;Relaxation theorems;419
6.1.1;9.1 Introduction;419
6.1.2;9.2 Relaxation Theorems;420
6.2;Implicit partial differential equations;442
6.2.1;10.1 Introduction;442
6.2.2;10.2 Existence theorems;443
6.2.3;10.3 Examples;454
6.3;Existence of minima for non- quasiconvex integrands;467
6.3.1;11.1 Introduction;467
6.3.2;11.2 Sufficient conditions;469
6.3.3;11.3 Necessary conditions;474
6.3.4;11.4 The scalar case;485
6.3.5;11.5 The vectorial case;489
7;Miscellaneous;502
7.1;Function spaces;503
7.1.1;12.1 Introduction;503
7.1.2;12.2 Main notation;503
7.1.3;12.3 Some properties of Hölder spaces;506
7.1.4;12.4 Some properties of Sobolev spaces;509
7.2;Singular values;514
7.2.1;13.1 Introduction;514
7.2.2;13.2 Definition and basic properties;514
7.2.3;13.3 Signed singular values and von Neumann type inequalities;518
7.3;Some underdetermined partial differential equations;527
7.3.1;14.1 Introduction;527
7.3.2;14.2 The equations div u = f and curl u = f;527
7.3.3;14.3 The equation det. u = f;533
7.4;Extension of Lipschitz functions on Banach spaces;546
7.4.1;15.1 Introduction;546
7.4.2;15.2 Preliminaries and notation;546
7.4.3;15.3 Norms induced by an inner product;548
7.4.4;15.4 Extension from a general subset of E to E;555
7.4.5;15.5 Extension from a convex subset of E to E;562
8;Bibliography;565
9;Notation;606
9.1;General notation;606
9.2;Convex analysis;606
9.3;Determinants and singular values;607
9.4;Quasiconvex analysis;609
9.5;Function spaces;609
10;Index;610