xii Contents
48.2. Uniqueness and parabolic regularization 235
48.3. The Cauchy problem on a finite time interval 237
48.4. Strictly hyperbolic equations of second order 240
§49. Domain of dependence 243
§50. Propagation of singularities 247
50.1. The null-bicharacteristics 247
50.2. Operators of real principal type 247
50.3. Propagation of singularities for operators of real principal
type 249
50.4. Propagation of singularities in the case of a hyperbolic
Cauchy problem 255
§51. Problems 258
Chapter VII. Elliptic Boundary Value Problems and Parametrices 263
Introduction to Chapter VII 263
§52. Pseudodifferential operators on a manifold 264
52.1. Manifolds and vector bundles 264
52.2. Definition of a pseudodifferential operator on a manifold 265
52.3. Elliptic ψdo’s on a manifold 266
§53. Boundary value problems in the half-space 266
53.1. Factorization of an elliptic symbol 266
53.2. Explicit solution of the boundary value problem 268
§54. Elliptic boundary value problems in a bounded domain 270
54.1. The method of “freezing” coefficients 270
54.2. The Fredholm property 273
54.3. Invariant form of the ellipticity of boundary conditions 276
54.4. Boundary value problems for elliptic systems of differential
equations 276
§55. Parametrices for elliptic boundary value problems 278
55.1. Plus-operators and minus-operators 278
55.2. Construction of the parametrix in the half-space 281
55.3. Parametrix in a bounded domain 284
§56. The heat trace asymptotics 285
56.1. The existence and the estimates of the resolvent 285
56.2. The parametrix construction 286
56.3. The heat trace for the Dirichlet Laplacian 288
56.4. The heat trace for the Neumann Laplacian 293
56.5. The heat trace for the elliptic operator of an arbitrary order 294
§57. Parametrix for the Dirichlet-to-Neumann operator 296
Previous Page Next Page