Partial Differential Equations II

Contents of Volumes I and III
Preface
7 Pseudodifferential Operators
1 The Fourier integral representation and symbol classes
2 Schwartz kernels of pseudodifferential operators
3 Adjoints and products
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Contents of Volumes I and III
Preface
7 Pseudodifferential Operators
1 The Fourier integral representation and symbol classes
2 Schwartz kernels of pseudodifferential operators
3 Adjoints and products
4 Elliptic operators and parametrices
5 L2-estimates
6 Gading's inequality
7 Hyperbolic evolution equations
8 Egorov's theorem
9 Microlocal regularity
10 Operators on manifolds
11 The method of layer potentials
12 Parametrix for regular elliptic boundary problems
13 Parametrix for the heat equation
14 The Weyl calculus
15 Operators of harmonic oscillator type
References
8 Spectral Theory
1 The spectral theorem
2 Self-adjoint differential operators
3 Heat asymptotics and eigenvalue asymptotics
4 The Laplace operator on Sn
5 The Laplace operator on hyperbolic space
6 The harmonic oscillator
7 The quantum Coulomb problem
8 The Laplace operator on cones
References
9 Scattering by Obstacles
1 The scattering problem
2 Eigenfunction expansions
3 The scattering operator
4 Connections with the wave equation
5 Wave operators
6 Translation representations and the Lax-Phillips semigroup Z(t)
7 Integral equations and scattering poles
8 Trace formulas; the scattering phase
9 Scattering by a sphere
10 Inverse problems I
11 Inverse problems lI
12 Scattering by rough obstacles
A Lidskii's trace theorem
References
10 Dirae Operators and Index Theory
1 Operators of Dirac type
2 Clifford algebras
3 Spinors
4 Weitzenbock formulas
5 Index of Dirac operators
6 Proof of the local index formula
7 The Chern-Gauss-Bonnet theorem
8 Spinc manifolds
9 The Riemann-Roch theorem
10 Direct attack in 2-D
11 Index of operators of harmonic oscillator type
References
11 Brownian Motion and Potential Theory
1 Brownian motion and Wiener measure
2 The Feynman-Kac formula
3 The Dirichlet problem and diffusion on domains with boundary
4 Martingales, stopping times, and the strong Markov property
5 First exit time and the Poisson integral
6 Newtonian capacity
7 Stochastic integrals
8 Stochastic integrals, II
9 Stochastic differential equations
10 Application to equations of diffusion
A The Trotter product formula
References
12 The a-Neumann Problem
A Elliptic complexes
1 The O-complex
2 Morrey's inequality, the Levi form, and strong pseudoconvexity
3 The 1-estimate and some consequences
4 Higher-order subelliptic estimates
5 Regularity via elliptic regularization
6 The Hodge decomposition and the 0-equation
7 The Bergman projection and Toeplitz operators
8 The a-Neumann problem on (O, q)-forms
9 Reduction to pseudodifferential equations on the boundary
10 The J-equation on complex manifolds and almost complex manifolds
B Complements on the Levi form
C The Neumann operator for the Dirichlet problem
References
C Connections and Curvature
1 Covariant derivatives and curvature on general vector bundles
2 Second covariant derivatives and covariant-exterior derivatives
3 The curvature tensor of a Riemannian manifold
4 Geometry of submanifolds and subbundles
5 The Gauss-Bonnet theorem for surfaces
6 The principal bundle picture
7 The Chern-Weil construction
8 The Chern--Gauss-Bonnet theorem
References
Index
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