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Direct Methods in the Calculus of Variations

:	Bernard Dacorogna
:	Direct Methods in the Calculus of Variations
:	Springer-Verlag
:	9780387552491
:	2
:	CHF 160.90
:

:	Analysis
:	English

:	622
:	Wasserzeichen
:	PC/MAC/eReader/Tablet
:	PDF

This book is developed for the study of vectorial problems in the calculus of variations. The subject is a very active one and almost half of the book consists of new material. This is a new edition of the earlier book published in 1989 and it is suitable for graduate students. The book has been updated with some new material and examples added. Applications are included.

	Contents	6
	Preface	12
	Introduction	14
	1.1 The direct methods of the calculus of variations	14
	1.2 Convex analysis and the scalar case	16
	1.3 Quasiconvex analysis and the vectorial case	22
	1.4 Relaxation and non-convex problems	30
	1.5 Miscellaneous	36
	Convex analysis and the scalar case	41
	Convex sets and convex functions	42
	2.1 Introduction	42
	2.2 Convex sets	43
	2.3 Convex functions	55
	Lower semicontinuity and existence theorems	83
	3.1 Introduction	83
	3.2 Weak lower semicontinuity	84
	3.3 Weak continuity and invariant integrals	111
	3.4 Existence theorems and Euler-Lagrange equations	115
	The one dimensional case	128
	4.1 Introduction	128
	4.2 An existence theorem	129
	4.3 The Euler-Lagrange equation	134
	4.4 Some inequalities	141
	4.5 Hamiltonian formulation	146
	4.6 Regularity	152
	4.7 Lavrentiev phenomenon	157
	Quasiconvex analysis and the vectorial case	161
	Polyconvex, quasiconvex and rank one convex functions	162
	5.1 Introduction	162
	5.2 Definitions and main properties	163
	5.3 Examples	185
	5.4 Appendix: some basic properties of determinants	256
	Polyconvex, quasiconvex and rank one convex envelopes	271
	6.1 Introduction	271
	6.2 The polyconvex envelope	272
	6.3 The quasiconvex envelope	277
	6.4 The rank one convex envelope	283
	6.5 Some more properties of the envelopes	286
	6.6 Examples	291
	Polyconvex, quasiconvex and rank one convex sets	319
	7.1 Introduction	319
	7.2 Polyconvex, quasiconvex and rank one convex sets	321
	7.3 The different types of convex hulls	329
	7.4 Examples	353
	Lower semi continuity and existence theorems in the vectorial case	373
	8.1 Introduction	373
	8.2 Weak lower semicontinuity	374
	8.3 Weak Continuity	399
	8.4 Existence theorems	409
	8.5 Appendix: some properties of Jacobians	413
	Relaxation and non-convex problems	418
	Relaxation theorems	419
	9.1 Introduction	419
	9.2 Relaxation Theorems	420
	Implicit partial differential equations	442
	10.1 Introduction	442
	10.2 Existence theorems	443
	10.3 Examples	454
	Existence of minima for non- quasiconvex integrands	467
	11.1 Introduction	467
	11.2 Sufficient conditions	469
	11.3 Necessary conditions	474
	11.4 The scalar case	485
	11.5 The vectorial case	489
	Miscellaneous	502
	Function spaces	503
	12.1 Introduction	503
	12.2 Main notation	503
	12.3 Some properties of Hölder spaces	506
	12.4 Some properties of Sobolev spaces	509
	Singular values	514
	13.1 Introduction	514
	13.2 Definition and basic properties	514
	13.3 Signed singular values and von Neumann type inequalities	518
	Some underdetermined partial differential equations	527
	14.1 Introduction	527
	14.2 The equations div u = f and curl u = f	527
	14.3 The equation det. u = f	533
	Extension of Lipschitz functions on Banach spaces	546
	15.1 Introduction	546
	15.2 Preliminaries and notation	546
	15.3 Norms induced by an inner product	548
	15.4 Extension from a general subset of E to E	555
	15.5 Extension from a convex subset of E to E	562
	Bibliography	565
	Notation	606
	General notation	606
	Convex analysis	606
	Determinants and singular values	607
	Quasiconvex analysis	609
	Function spaces	609
	Index	610