Calculus of Variations

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
Exercises......Page 47
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
Exercises......Page 77
3.1 A finite dimensional example......Page 79
3.2 The construction of Lyusternik-Schnirelman......Page 84
Exercises......Page 95
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
Exercises......Page 119
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
Exercises......Page 174
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 Poincare and Sobolev inequalities......Page 192
Exercises......Page 198
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
Exercises......Page 220
5.1 Nonlower semicontinuous functionals and relaxation......Page 222
5.2 Representation of relaxed functionals via convex envelopes......Page 230
Exercises......Page 241
6.1 The definition of Γ-convergence......Page 242
6.2 Homogenization......Page 248
6.3 Thin insulating layers......Page 252
Exercises......Page 257
7.1 The space BV(Ω)......Page 258
7.2 The example of Modica-Mortola......Page 265
Exercises......Page 273
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
Exercises......Page 306
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
Exercises......Page 335
Ιndex......Page 336