Generalized Solutions of Hamilton-Jacobi Equations

Contents......Page 4
Introduction......Page 7
Part 1: Generalities......Page 15
1.1 Notations......Page 16
1.2 Classical methods: characteristics......Page 18
1.3 Optimal Control theory......Page 27
1.4 The vanishing viscosity method......Page 49
1.5 Viscosity solutions: uniqueness and stability......Page 53
1.6 Accretivity of the Hamilton-Jacobi operator......Page 64
Part 2: The Dirichlet Problem......Page 68
2.1 The main existence result......Page 69
2.2 The case when \Omega is bounded and smooth, and H is superquadratic......Page 71
2.3 The general case with \Omega bounded......Page 76
2.4 The general case with \Omega unbounded......Page 82
3.1 Uniqueness and stability results for SSH solutions......Page 87
3.2 A Lemma......Page 95
3.3 Application to some regularity results......Page 97
3.4 Relations with viscosity solutions......Page 101
4.1 The case when \Omega = R^N......Page 104
4.2 The general case......Page 107
4.3 A geometrical assumption......Page 112
4.5 Uniqueness results......Page 117
5.1 Introduction......Page 121
5.2 The case of the Hamiltonian: H(x,p) = |p|-n(x)......Page 122
5.3 The general case of a convex Hamiltonian......Page 131
5.4 Extensions......Page 141
5.5 Application to the classification of solutions in a degenerate case......Page 148
6.1 Incompatible boundary conditions and singular perturbations......Page 153
6.2 The rate of convergence of the vanishing viscosity method......Page 161
7.1 A result on the existence of classical solutions......Page 167
7.2 Maximum subsolutions in the convex case......Page 170
8.1 Neumann boundary conditions for Hamilton-Jacobi equations......Page 175
8.2 The obstacle problem for Hamilton-Jacobi equations......Page 181
8.3 Regularity of solutions near the boundary......Page 183
8.4 Optimal control theory and Hamilton-Jacobi equations......Page 195
8.5 Various questions......Page 206
Part 3: The Cauchy Problem......Page 207
9.1 Introduction......Page 208
9.2 Main existence results......Page 209
10.1 Uniqueness for SSH solutions in the case of a convex Hamiltonian......Page 214
10.2 Uniqueness in the general case......Page 215
10.3 Relations with viscosity solutions......Page 218
11.1 Compatibility conditions and Lax formula......Page 222
11.2 Some extensions......Page 227
11.3 Singular4 perturbations and the vanishing viscosity method......Page 234
12.1 Classical solutions......Page 238
12.2 Weak solutions......Page 240
13.1 Regularizing effect in R^N......Page 243
13.2 Boundary conditions......Page 248
14.1 Localization: the domain of dependence......Page 252
14.2 Asymptotics......Page 256
15.1 The threshold of regularity......Page 261
15.2 Regularity of solutions near the boundary......Page 266
15.3 Various questions......Page 269
16.1 Applications to some hyperbolic systems......Page 273
16.2 Singular perturbations and large-scale systems......Page 276
16.3 Asymptotic problems......Page 278
16.4 Various questions......Page 283
Appendix 1: Existence and a priori bounds for solutions of second order quasilinear equations......Page 285
Appendix 2: A few results on viscosity solutions......Page 292
References......Page 314