偏微分方程 第2卷 第2版 英文2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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7 Pseudodifferential Operators1

1 The Fourier integral representation and symbol classes2

2 Schwartz kernels of pseudodifferential operators5

3 Adjoints and products10

4 Elliptic operators and parametrices15

5 L2-estimates18

6 G？rding's inequality22

7 Hyperbolic evolution equations23

8 Egorov's theorem26

9 Microlocal regularity29

10 Operators on manifolds33

11 The method of layer potentials36

12 Parametrix for regular elliptic boundary problems47

13 Parametrix for the heat equation56

14 The Weyl calculus67

15 Operators of harmonic oscillator type80

References88

8 Spectral Theory91

1 The spectral theorem92

2 Self-adjoint differential operators100

3 Heat asymptotics and eigenvalue asymptotics106

4 The Laplace operator on Sn113

5 The Laplace operator on hyperbolic space123

6 The harmonic oscillator126

7 The quantum Coulomb problem135

8 The Laplace operator on cones149

References171

9 Scattering by Obstacles175

1 The scattering problem177

2 Eigenfunction expansions186

3 The scattering operator192

4 Connections with the wave equation197

5 Wave operators205

6 Translation representations and the Lax-Phillips semigroup Z(t)211

7 Integral equations and scattering poles218

8 Trace formulas;the scattering phase232

9 Scattering by a sphere239

10 Inverse problemsⅠ248

11 InverseproblemsⅡ254

12 Scattering by rough obstacles266

A Lidskii's trace theorem275

References277

10 Dirac Operators and Index Theory281

1 Operators of Dirac type283

2 Clifford algebras289

3 Spinors294

4 Weitzenbock formulas300

5 Index of Dirac operators306

6 Proof of the local index formula309

7 The Chern-Gauss-Bonnet theorem316

8 Spinc manifolds320

9 The Riemann-Roch theorem325

10 Direct attack in 2-D338

11 Index of operators of harmonic oscillator type345

References358

11 Brownian Motion and Potential Theory361

1 Brownian motion and Wiener measure363

2 The Feynman-Kac formula370

3 The Dirichlet problem and diffusion on domains with boundary375

4 Martingales,stopping times,and the strong Markov property384

5 First exit time and the Poisson integral394

6 Newtonian capacity398

7 Stochastic integrals412

8 Stochastic integrals,Ⅱ423

9 Stochastic differential equations430

10 Application to equations of diffusion437

A The Trotter product formula448

References454

12 The ？-Neumann Problem457

A Elliptic complexes460

1 The ？-complex465

2 Morrey's inequality,the Levi form,and strong pseudoconvexity469

3 The 1/2-estimate and some consequences472

4 Higher-order subelliptic estimates476

5 Regularity viaelliptic regularization480

6 The Hodge decomposition and the ？-equation483

7 The Bergman projection and Toeplitz operators487

8 The ？-Neumann problem on（0,q）-forms494

9 Reduction to pseudodifferential equations on the boundary503

10 The ？-equation on complex manifolds and almost complex manifolds516

B Complements on the Levi form527

C The Neumann operator for the Dirichlet problem531

References535

C Connections and Curvature539

1 Covariant derivatives and curvature on general vector bundles540

2 Second covariant derivatives and covariant-exterior derivatives546

3 The curvature tensor of a Riemannian manifold548

4 Geometry of submanifolds and subbundles560

5 The Gauss-Bonnet theorem for surfaces574

6 The principal bundle picture586

7 The Chern-Weil construction594

8 The Chern-Gauss-Bonnet theorem598

References608

Index611