Calculus of Variations and Partial
          Differential Equations
           Diogo Aguiar Gomes
                                                                                     Contents
                                        .   Introduction                                                                                             5
                                         1.    Finite dimensional optimization problems                                                              9
                                                                                                       n
                                           1.     Unconstrained minimization in R                                                                  10
                                           2.     Convexity                                                                                        16
                                           3.     Lagrange multipliers                                                                             26
                                           4.     Linear programming                                                                               30
                                           5.     Non-linear optimization with constraints                                                         37
                                           6.     Bibliographical notes                                                                            48
                                         2.    Calculus of variations in one independent variable                                                  49
                                           1.     Euler-Lagrange Equations                                                                         50
                                           2.     Further necessary conditions                                                                     57
                                           3.     Applications to Riemannian geometry                                                              60
                                           4.     Hamiltonian dynamics                                                                             75
                                           5.     Sufficient conditions                                                                              89
                                           6.     Symmetries and Noether theorem                                                                 105
                                           7.     Critical point theory                                                                          111
                                           8.     Invariant measures                                                                             116
                                           9.     Non convex problems                                                                            118
                                           10.      Geometry of Hamiltonian systems                                                              119
                                           11.      Perturbation theory                                                                          122
                                           12.      Bibliographical notes                                                                        126
                                         3.    Calculus of variations and elliptic equations                                                     127
                                           1.     Euler-Lagrange equation                                                                        129
                                           2.     Further necessary conditions and applications                                                  136
                                           3.     Convexity and sufficient conditions                                                              136
                                           4.     Direct method in the calculus of variations                                                    136
                                                                                              3
                            4                                 CONTENTS
                               5.   Euler-Lagrange equations                                            145
                               6.   Regularity by energy methods                                        146
                               7.   H¨older continuity                                                  155
                               8.   Schauder estimates                                                  171
                             4.   Optimal control and viscosity solutions                               183
                               1.   Elementary examples and properties                                  186
                               2.   Dynamic programming principle                                       188
                               3.   Pontryagin maximum principle                                        190
                               4.   The Hamilton-Jacobi equation                                        192
                               5.   Verification theorem                                                 193
                               6.   Existence of optimal controls - bounded control space               195
                               7.   Sub and superdifferentials                                           197
                               8.   Optimal control in the calculus of variations setting               202
                               9.   Viscosity solutions                                                 214
                               10.   Stationary problems                                                224
                             5.   Duality theory                                                        231
                               1.   Model problems                                                      231
                               2.   Some informal computations                                          237
                               3.   Duality                                                             241
                               4.   Generalized Mather problem                                          244
                               5.   Monge-Kantorowich problem                                           266
                            .   Bibliography                                                            269
                            .   Index                                                                   271