Calculus of Variations

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Overview

This textbook on the calculus of variations leads the reader from the basics to modern aspects of the theory. One-dimensional problems and the classical issues such as Euler-Lagrange equations are treated, as are Noether's theorem, Hamilton-Jacobi theory, and in particular geodesic lines, thereby developing some important geometric and topological aspects. The second part of the book deals with multiple integrals. After a review of Lebesgue integration, Banach and Hilbert space theory and Sobolev spaces (with complete and detailed proofs), there is a treatment of the direct methods and the fundamental lower semicontinuity theorems. Subsequent chapters introduce the basic concepts of the modern calculus of variations, namely relaxation, Gamma convergence, bifurcation theory and minimax methods based on the Palais-Smale condition. After having studied this book, the reader will be well equipped to read research papers in the calculus of variations.
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Editorial Reviews

From the Publisher
"This modern self-contained exposition...is an excellent textbook for graduate students and a good source of information in the calculus of variations." Mathematical Reviews
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Product Details

Table of Contents

Preface and summary
Remarks on notation
Pt. 1 One-dimensional variational problems 1
1 The classical theory 3
2 A geometric example: geodesic curves 32
3 Saddle point constructions 62
4 The theory of Hamilton and Jacobi 79
5 Dynamic optimization 104
Pt. 2 Multiple integrals in the calculus of variations 115
1 Lebesgue measure and integration theory 117
2 Banach spaces 125
3 L[superscript p] and Sobolev spaces 159
4 The direct methods in the calculus of variations 183
5 Nonconvex functionals. Relaxation 205
6 [Gamma]-convergence 225
7 BV-functionals and [Gamma]-convergence: the example of Modica and Mortola 241
App. A The coarea formula 257
App. B The distance function from smooth hypersurfaces 262
8 Bifurcation theory 266
9 The Palais-Smale condition and unstable critical points of variational problems 291
Index 319
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