UID:
almafu_9959239099802883
Umfang:
1 online resource (xii, 382 pages) :
,
digital, PDF file(s).
ISBN:
1-139-88631-2
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1-107-36649-6
,
1-107-37120-1
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1-107-36158-3
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1-107-36811-1
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1-299-40427-8
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1-107-36403-5
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0-511-56942-4
Serie:
London Mathematical Society lecture note series ; 202
Inhalt:
Pseudodifferential operators arise naturally in the solution of boundary problems for partial differential equations. The formalism of these operators serves to make the Fourier-Laplace method applicable for nonconstant coefficient equations. This book presents the technique of pseudodifferential operators and its applications, especially to the Dirac theory of quantum mechanics. The treatment uses 'Leibniz formulas' with integral remainders or as asymptotic series. A pseudodifferential operator may also be described by invariance under action of a Lie-group. The author discusses connections to the theory of C*-algebras, invariant algebras of pseudodifferential operators under hyperbolic evolution and the relation of the hyperbolic theory to the propagation of maximal ideals. This book will be of particular interest to researchers in partial differential equations and mathematical physics.
Anmerkung:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
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Cover; Title; Copyright; Dedication; Table of Contents; Chapter 0. Introductory discussions; 0.0. Some special notations, used in the book; 0.1. The Fourier transform; elementary facts; 0.2. Fourier analysis for temperate distributions on Rn; 0.3. The Paley-Wiener theorem; Fourier transform forgeneral uЄ D'; 0.4. The Fourier-Laplace method; examples; 0.5. Abstract solutions and hypo-ellipticity; 0.6. Exponentiating a first order partial differential operator.; 0.7. Solving a nonlinear first order partial differential equation; 0.8.Characteristics and bicharacteristics of a linear PDE
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0.9. Lie groups and Lie algebras, for classical analystsChapter 1. Calculus of pseudodifferential operators; 1.0. Introduction; 1.1. Definition of ψdo's; 1.2. Elementary properties of ψdo's; 1.3. Hoermander symbols; Wey ψdo' s; distribution kernels; 1.4. The composition formulas of Beals; 1.5. The Leibniz' formulas with integral remainder; 1.6. Calculus of ψdo's for symbols of Hoermander type; 1.7. Strictly classical symbols; some lemmata for application; Chapter 2. Elliptic operators and parametrices in Rn; 2.0. Introduction; 2.1. Elliptic and md-elliptic ψdo's
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2.2. Formally hypo-elliptic ψdo's2.3. Local md-ellipticity and local md-hypo-ellipticity; 2.4. Formally hypo-elliptic differential expressions; 2.5. The wave front set and its invariance under ψdo's; 2.6. Systems of ψdo's; Chapter 3. L2-Sobolev theory and applications; 3.0. Introduction; 3.1. L2-boundedness of zero-order ψdo's; 3.2. L2-boundedness for the case of δ〉0; 3.3. Weighted Sobolev spaces; K-parametrix and Green inverse; 3.4. Existence of a Green inverse; 3.5. Hs-compactness for ψdo's of negative order; Chapter 4. Pseudodifferential operators on manifolds with conical ends
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5.5. A system of ψde's for the ψj of problem 3.45.6. Lopatinskij-Shapiro conditions; normal solvability of (2.2).; 5.7. Hypo-ellipticity, and the classical parabolic problem; 5.8. Spectral- and semigroup theory for ψdo's; 5.9. Self-adjointness for boundary problems; 5.10. C*-algebras of ψdo's; comparison algebras; Chapter 6. Hyperbolic first order systems; 6.0.Introduction; 6.1. First order symmetric hyperbolic systems of PDE; 6.2. First order symmetric hyperbolic systems of ψde's on Rn .; 6.3. The evolution operator and its properties
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6.4. N-th order strictly hyperbolic systems andsymmetrizers.
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English
Weitere Ausg.:
ISBN 0-521-37864-8
Sprache:
Englisch
URL:
https://doi.org/10.1017/CBO9780511569425
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