Partial Differential Equations II

1 - Contents [Seite 8]
2 - Contents of Volumes I and III [Seite 12]
3 - Preface [Seite 14]
4 - 7 Pseudodifferential Operators [Seite 24]
4.1 - 1 The Fourier integral representation and symbol classes [Seite 25]
4.2 - 2 Schwartz kernels of pseudodifferential operators [Seite 28]
4.3 - 3 Adjoints and products [Seite 33]
4.4 - 4 Elliptic operators and parametrices [Seite 38]
4.5 - 5 L2-estimates [Seite 41]
4.6 - 6 Gårding's inequality [Seite 45]
4.7 - 7 Hyperbolic evolution equations [Seite 46]
4.8 - 8 Egorov's theorem [Seite 49]
4.9 - 9 Microlocal regularity [Seite 52]
4.10 - 10 Operators on manifolds [Seite 56]
4.11 - 11 The method of layer potentials [Seite 59]
4.12 - 12 Parametrix for regular elliptic boundary problems [Seite 70]
4.13 - 13 Parametrix for the heat equation [Seite 79]
4.14 - 14 The Weyl calculus [Seite 90]
4.15 - 15 Operators of harmonic oscillator type [Seite 103]
4.16 - References [Seite 111]
5 - 8 Spectral Theory [Seite 114]
5.1 - 1 The spectral theorem [Seite 115]
5.2 - 2 Self-adjoint differential operators [Seite 123]
5.3 - 3 Heat asymptotics and eigenvalue asymptotics [Seite 129]
5.4 - 4 The Laplace operator on Sn [Seite 136]
5.5 - 5 The Laplace operator on hyperbolic space [Seite 146]
5.6 - 6 The harmonic oscillator [Seite 149]
5.7 - 7 The quantum Coulomb problem [Seite 158]
5.8 - 8 The Laplace operator on cones [Seite 172]
5.9 - References [Seite 194]
6 - 9 Scattering by Obstacles [Seite 197]
6.1 - 1 The scattering problem [Seite 199]
6.2 - 2 Eigenfunction expansions [Seite 208]
6.3 - 3 The scattering operator [Seite 214]
6.4 - 4 Connections with the wave equation [Seite 219]
6.5 - 5 Wave operators [Seite 227]
6.6 - 6 Translation representations and the Lax-Phillips semigroup Z(t) [Seite 233]
6.7 - 7 Integral equations and scattering poles [Seite 240]
6.8 - 8 Trace formulas [Seite 254]
6.9 - 9 Scattering by a sphere [Seite 261]
6.10 - 10 Inverse problems I [Seite 270]
6.11 - 11 Inverse problems II [Seite 276]
6.12 - 12 Scattering by rough obstacles [Seite 288]
6.13 - A Lidskii's trace theorem [Seite 297]
6.14 - References [Seite 299]
7 - 10 Dirac Operators and Index Theory [Seite 303]
7.1 - 1 Operators of Dirac type [Seite 305]
7.2 - 2 Clifford algebras [Seite 311]
7.3 - 3 Spinors [Seite 316]
7.4 - 4 Weitzenbock formulas [Seite 322]
7.5 - 5 Index of Dirac operators [Seite 328]
7.6 - 6 Proof of the local index formula [Seite 331]
7.7 - 7 The Chern-Gauss-Bonnet theorem [Seite 338]
7.8 - 8 Spinc manifolds [Seite 342]
7.9 - 9 The Riemann-Roch theorem [Seite 347]
7.10 - 10 Direct attack in 2-D [Seite 360]
7.11 - 11 Index of operators of harmonic oscillator type [Seite 367]
7.12 - References [Seite 380]
8 - 11 Brownian Motion and Potential Theory [Seite 383]
8.1 - 1 Brownian motion and Wiener measure [Seite 385]
8.2 - 2 The Feynman-Kac formula [Seite 392]
8.3 - 3 The Dirichlet problem and diffusion on domains with boundary [Seite 397]
8.4 - 4 Martingales, stopping times, and the strong Markov property [Seite 406]
8.5 - 5 First exit time and the Poisson integral [Seite 416]
8.6 - 6 Newtonian capacity [Seite 420]
8.7 - 7 Stochastic integrals [Seite 434]
8.8 - 8 Stochastic integrals, II [Seite 445]
8.9 - 9 Stochastic differential equations [Seite 452]
8.10 - 10 Application to equations of diffusion [Seite 459]
8.11 - A The Trotter product formula [Seite 470]
8.12 - References [Seite 476]
9 - 12 The -Neumann Problem [Seite 479]
9.1 - A Elliptic complexes [Seite 482]
9.2 - 1 The -complex [Seite 487]
9.3 - 2 Morrey's inequality, the Levi form, and strong pseudoconvexity [Seite 491]
9.4 - 3 The 1/2-estimate and some consequences [Seite 494]
9.5 - 4 Higher-order subelliptic estimates [Seite 498]
9.6 - 5 Regularity via elliptic regularization [Seite 502]
9.7 - 6 The Hodge decomposition and the -equation [Seite 505]
9.8 - 7 The Bergman projection and Toeplitz operators [Seite 509]
9.9 - 8 The -Neumann problem on (0,q)-forms [Seite 516]
9.10 - 9 Reduction to pseudodifferential equations on the boundary [Seite 525]
9.11 - 10 The -equation on complex manifolds and almost complex manifolds [Seite 538]
9.12 - B Complements on the Levi form [Seite 549]
9.13 - C The Neumann operator for the Dirichlet problem [Seite 553]
9.14 - References [Seite 557]
10 - C Connections and Curvature [Seite 560]
10.1 - 1 Covariant derivatives and curvature on general vector bundles [Seite 561]
10.2 - 2 Second covariant derivatives and covariant-exterior derivatives [Seite 567]
10.3 - 3 The curvature tensor of a Riemannian manifold [Seite 569]
10.4 - 4 Geometry of submanifolds and subbundles [Seite 581]
10.5 - 5 The Gauss-Bonnet theorem for surfaces [Seite 595]
10.6 - 6 The principal bundle picture [Seite 607]
10.7 - 7 The Chern-Weil construction [Seite 615]
10.8 - 8 The Chern-Gauss-Bonnet theorem [Seite 619]
10.9 - References [Seite 629]
11 - Index [Seite 631]