Partial Differential Equations II (eBook)

Michael E. Taylor is a Professor at University of North Carolina in the Department of Mathematics.

Contents 8
Contents of Volumes I and III 12
Preface 14
7 Pseudodifferential Operators 24
1 The Fourier integral representation and symbol classes 25
2 Schwartz kernels of pseudodifferential operators 28
3 Adjoints and products 33
4 Elliptic operators and parametrices 38
5 L2-estimates 41
6 Gårding's inequality 45
7 Hyperbolic evolution equations 46
8 Egorov's theorem 49
9 Microlocal regularity 52
10 Operators on manifolds 56
11 The method of layer potentials 59
12 Parametrix for regular elliptic boundary problems 70
13 Parametrix for the heat equation 79
14 The Weyl calculus 90
15 Operators of harmonic oscillator type 103
References 111
8 Spectral Theory 114
1 The spectral theorem 115
2 Self-adjoint differential operators 123
3 Heat asymptotics and eigenvalue asymptotics 129
4 The Laplace operator on Sn 136
5 The Laplace operator on hyperbolic space 146
6 The harmonic oscillator 149
7 The quantum Coulomb problem 158
8 The Laplace operator on cones 172
References 194
9 Scattering by Obstacles 197
1 The scattering problem 199
2 Eigenfunction expansions 208
3 The scattering operator 214
4 Connections with the wave equation 219
5 Wave operators 227
6 Translation representations and the Lax–Phillips semigroup Z(t) 233
7 Integral equations and scattering poles 240
8 Trace formulas the scattering phase
9 Scattering by a sphere 261
10 Inverse problems I 270
11 Inverse problems II 276
12 Scattering by rough obstacles 288
A Lidskii's trace theorem 297
References 299
10 Dirac Operators and Index Theory 303
1 Operators of Dirac type 305
2 Clifford algebras 311
3 Spinors 316
4 Weitzenbock formulas 322
5 Index of Dirac operators 328
6 Proof of the local index formula 331
7 The Chern–Gauss–Bonnet theorem 338
8 Spinc manifolds 342
9 The Riemann–Roch theorem 347
10 Direct attack in 2-D 360
11 Index of operators of harmonic oscillator type 367
References 380
11 Brownian Motion and Potential Theory 383
1 Brownian motion and Wiener measure 385
2 The Feynman–Kac formula 392
3 The Dirichlet problem and diffusion on domains with boundary 397
4 Martingales, stopping times, and the strong Markov property 406
5 First exit time and the Poisson integral 416
6 Newtonian capacity 420
7 Stochastic integrals 434
8 Stochastic integrals, II 445
9 Stochastic differential equations 452
10 Application to equations of diffusion 459
A The Trotter product formula 470
References 476
12 The -Neumann Problem 479
A Elliptic complexes 482
1 The -complex 487
2 Morrey's inequality, the Levi form, and strong pseudoconvexity 491
3 The 1/2-estimate and some consequences 494
4 Higher-order subelliptic estimates 498
5 Regularity via elliptic regularization 502
6 The Hodge decomposition and the -equation 505
7 The Bergman projection and Toeplitz operators 509
8 The -Neumann problem on (0,q)-forms 516
9 Reduction to pseudodifferential equations on the boundary 525
10 The -equation on complex manifolds and almost complex manifolds 538
B Complements on the Levi form 549
C The Neumann operator for the Dirichlet problem 553
References 557
C Connections and Curvature 560
1 Covariant derivatives and curvature on general vector bundles 561
2 Second covariant derivatives and covariant-exterior derivatives 567
3 The curvature tensor of a Riemannian manifold 569
4 Geometry of submanifolds and subbundles 581
5 The Gauss–Bonnet theorem for surfaces 595
6 The principal bundle picture 607
7 The Chern–Weil construction 615
8 The Chern–Gauss–Bonnet theorem 619
References 629
Index 631