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Partial Differential Equations II Qualitative Studies of Linear Equations

:	Michael Taylor
:	Partial Differential Equations II Qualitative Studies of Linear Equations
:	Springer-Verlag
:	9781441970527
:	2
:	CHF 188.20
:

:	Analysis
:	English

:	614
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

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.

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