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 basic ideas of optimal control theory are also given. 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. The prerequisites are knowledge of the basic results from calculus of one and several variables. After having studied this book, the reader will be well equipped to read research papers in the calculus of variations.
Author(s): Jürgen Jost, Xianqing Li-Jost
Series: Cambridge Studies in Advanced Mathematics 64
Publisher: Cambridge University Press
Year: 1999
Language: English
Pages: 340
Cover......Page 1
Title: Calculus of variations......Page 4
Copyright......Page 5
Contents......Page 8
Preface and summary......Page 11
Remarks on notation......Page 16
Part one: One-dimensional variational problems......Page 18
1.1 The Euler-Lagrange equations. Examples......Page 20
1.2 The idea of the direct methods and some regularity results......Page 27
1.3 The second variation. Jacobi fields......Page 35
1.4 Free boundary conditions......Page 41
1.5 Symmetries and the theorem of E. Noether......Page 43
2.1 The length and energy of curves......Page 49
2.2 Fields of geodesic curves......Page 60
2.3 The existence of geodesies......Page 68
3.1 A finite dimensional example......Page 79
3.2 The construction of Lyusternik-Schnirelman......Page 84
4.1 The canonical equations......Page 96
4.2 The Hamilton-Jacobi equation......Page 98
4.3 Geodesies......Page 104
4.4 Fields of extremals......Page 106
4.5 Hilbert's invariant integral and Jacobi's theorem......Page 109
4.6 Canonical transformations......Page 112
5.1 Discrete control problems......Page 121
5.2 Continuous control problems......Page 123
5.3 The Pontryagin maximum principle......Page 126
Part two: Multiple integrals in the calculus of variations......Page 132
1.1 The Lebesgue measure and the Lebesgue integral......Page 134
1.2 Convergence theorems......Page 139
2.1 Definition and basic properties of Banach and Hilbert spaces......Page 142
2.2 Dual spaces and weak convergence......Page 149
2.3 Linear operators between Banach spaces......Page 161
2.4 Calculus in Banach spaces......Page 167
3.1 L^p spaces......Page 176
3.2 Approximation of L^p functions by smooth functions (mollification)......Page 183
3.3 Sobolev spaces......Page 188
3.4 Rellich's theorem and the Poincaré and Sobolev inequalities......Page 192
4.1 Description of the problem and its solution......Page 200
4.2 Lower semicontinuity......Page 201
4.3 The existence of minimizers for convex variational problems......Page 204
4.4 Convex functionals on Hilbert spaces and Moreau-Yosida approximation......Page 207
4.5 The Euler-Lagrange equations and regularity questions......Page 212
5.1 Nonlower semicontinuous functionals and relaxation......Page 222
5.2 Representation of relaxed functionals via convex envelopes......Page 230
6.1 The definition of Γ-convergence......Page 242
6.2 Homogenization......Page 248
6.3 Thin insulating layers......Page 252
7.1 The space BV(Ω)......Page 258
7.2 The example of Modica-Mortola......Page 265
Appendix A The coarea formula......Page 274
Appendix B The distance function from smooth hypersurfaces......Page 279
8.1 Bifurcation problems in the calculus of variations......Page 283
8.2 The functional analytic approach to bifurcation theory......Page 287
8.3 The existence of catenoids as an example of a bifurcation process......Page 299
9.1 The Palais-Smale condition......Page 308
9.2 The mountain pass theorem......Page 318
9.3 Topological indices and critical points......Page 323
Index......Page 336
