偏微分方程. 第2卷 : 第2版

Contents of Volumes I and III

Preface

7 Pseudo differential Operators

1 The Fourier integral representation and symbol classes

2 Schwartz kernels of pseudo differential operators

3 Adjoints and products

4 Elliptic operators and parametrices

5 L2 estimates

6 Garding'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 Wey l calculus

15 Operators of harmonic oscillator type

Reference

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 equatio

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 l

11 Inverse problems II

12 Scattering by rough obstacles

A Lidskii's trace theorem

References

10 Dirac 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 diffusio

A The Trotter product formula

References

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This work is intended to provide a course of study of some of the major aspects of PDE. It is addressed to readers with a background in the basic introductory graduate mathematics courses in American universities: elementary real and complex analysis， differential geometry， and measure theory.