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Degenerate Elliptic Equations : NATO Asi Series - Serge Levendorskii

Degenerate Elliptic Equations

NATO Asi Series

Hardcover

Published: 1st September 1993
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0.1 The partial differential equation (1) (Au)(x) = L aa(x)(Dau)(x) = f(x) m lal9 is called elliptic on a set G, provided that the principal symbol a2m(X,) = L aa(x)a lal=2m of the operator A is invertible on G X (~n 0); A is called elliptic on G, too. This definition works for systems of equations, for classical pseudo differential operators ("pdo), and for operators on a manifold n. Let us recall some facts concerning elliptic operators. 1 If n is closed, then for any s E ~ , is Fredholm and the following a priori estimate holds (2) 1 2 Introduction If m > 0 and A : C=(O; C') -+ L (0; C') is formally self - adjoint 2 with respect to a smooth positive density, then the closure Ao of A is a self - adjoint operator with discrete spectrum and for the distribu­ tion functions of the positive and negative eigenvalues (counted with multiplicity) of Ao one has the following Weyl formula: as t -+ 00, (3) n 2m = t / II N±(1,a2m(x,e))dxde T·OO (on the right hand side, N±(t,a2m(x,e))are the distribution functions of the matrix a2m(X,e) : C' -+ CU).

Introductionp. 1
General Calculus of Pseudodifferential Operatorsp. 9
Weyl-Hormander Calculusp. 9
The Calculus of Pseudodifferential Operators with Double Symbolsp. 45
Model Classes of Degenerate Elliptic Differential Operatorsp. 75
Classes of Operators and Weighted Sobolev Spacesp. 75
Operators of Type 1 (Strong Degeneration Case)p. 81
Operators of Type 2 (Ellipticity along Boundary and Strong Degeneration in Normal Direction)p. 87
Operators of Type 3 (Ellipticity along Boundary and Euler Operators in Normal Direction)p. 95
Operators of Type 4 (Equations which Require Boundary and Coboundary Conditions)p. 98
General Classes of Degenerate Elliptic Differential Operatorsp. 129
Definition of Types of Operators and their Symbolsp. 129
Operators of Type 1p. 135
Operators of Type 4p. 145
Operators of Types 2, 3p. 158
Degenerate Elliptic Operators in Non-Power-Like Degeneration Casep. 163
Operators of Type 1 - 3p. 163
Operators of Type 4p. 165
L[subscript p]-Theory for Degenerate Elliptic Operatorsp. 171
L[subscript p]-Theory for Pseudodifferential Operators with Double Symbolsp. 171
Operators of Type 1p. 180
Operators of Type 4p. 182
Coersiveness of Degenerate Quadratic Formsp. 187
Types of Degenerate Quadratic Forms and their Symbolsp. 187
Forms of Type 1p. 190
Forms of Type 4p. 193
Forms of Types 2, 3p. 199
Forms of Type 3p. 202
Some Classes of Hypoelliptic Pseudodifferential Operators on Closed Manifoldp. 203
Operators of Slowly Varying Orderp. 203
Hypoelliptic Operators with Multiple Characteristicsp. 213
Weighted Sobolev Spaces and Hypoelliptic Operators with Multiple Characteristics as Fredholm Operatorsp. 225
Interior Boundary Value Problemp. 239
Algebra of Boundary Value Problems for Class of Pseudodifferential Operators which Change Order on the Boundaryp. 245
Symbols on [actual symbol not reproducible]p. 245
Classes of operators on Half-Spacep. 251
Weighted Sobolev Spacesp. 262
Operators on Closed Manifolds with Boundariesp. 270
An Index Theoremp. 275
General Schemes of Investigation of Spectral Asymptotics for Degenerate Elliptic Equationsp. 279
General Theorems on Spectral Asymptoticsp. 279
General Schemes of Investigation of Spectral Asymptotics and Generalizations of the Weyl Formula for Degenerate Elliptic Operatorsp. 291
Spectral Asymptotics of Degenerate Elliptic Operatorsp. 301
Formal Computations of Spectral Asymptotics for Operators of All Typesp. 301
Proof of the Asymptotic Formulaep. 317
Spectral Asymptotics of Hypoelliptic Operators with Multiple Characteristicsp. 335
Formal Computations of Spectral Asymptoticsp. 335
Proofs of the Asymptotic Formulaep. 355
A Brief Review of the Bibliographyp. 389
Bibliographyp. 399
Index of Notationp. 423
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792323051
ISBN-10: 079232305X
Series: NATO Asi Series
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 436
Published: 1st September 1993
Publisher: SPRINGER VERLAG GMBH
Country of Publication: NL
Dimensions (cm): 23.39 x 15.6  x 2.54
Weight (kg): 0.81