Taylor Partial Differential Equations II

Qualitative Studies of Linear Equations
2. Auflage 2011
ISBN: 978-1-4419-7052-7
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 614 Seiten

Reihe: Applied Mathematical Sciences

This second in the series of three volumes builds upon the basic theory of linear PDE given in volume 1, and pursues more advanced topics. Analytical tools introduced here include pseudodifferential operators, the functional analysis of self-adjoint operators, and Wiener measure. The book also develops basic differential geometrical concepts, centred about curvature. Topics covered include spectral theory of elliptic differential operators, the theory of scattering of waves by obstacles, index theory for Dirac operators, and Brownian motion and diffusion.

Zielgruppe

Research

Autoren/Hrsg.

Taylor, Michael E.

Inhaltsverzeichnis

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

Über Autor(innen)

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