Degenerate Elliptic Equations

By: Serge Levendorskii

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Hardcover | 30 June 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,â¬) = L aa(x)â¬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).

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Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Science Physics Quantum Physics & Quantum Mechanics & Quantum Field Theory Engineering & Technology Mechanical Engineering & Materials Mechanical Engineering

Introduction	p. 1
General Calculus of Pseudodifferential Operators	p. 9
Weyl-Hormander Calculus	p. 9
The Calculus of Pseudodifferential Operators with Double Symbols	p. 45
Model Classes of Degenerate Elliptic Differential Operators	p. 75
Classes of Operators and Weighted Sobolev Spaces	p. 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 Operators	p. 129
Definition of Types of Operators and their Symbols	p. 129
Operators of Type 1	p. 135
Operators of Type 4	p. 145
Operators of Types 2, 3	p. 158
Degenerate Elliptic Operators in Non-Power-Like Degeneration Case	p. 163
Operators of Type 1 - 3	p. 163
Operators of Type 4	p. 165
L[subscript p]-Theory for Degenerate Elliptic Operators	p. 171
L[subscript p]-Theory for Pseudodifferential Operators with Double Symbols	p. 171
Operators of Type 1	p. 180
Operators of Type 4	p. 182
Coersiveness of Degenerate Quadratic Forms	p. 187
Types of Degenerate Quadratic Forms and their Symbols	p. 187
Forms of Type 1	p. 190
Forms of Type 4	p. 193
Forms of Types 2, 3	p. 199
Forms of Type 3	p. 202
Some Classes of Hypoelliptic Pseudodifferential Operators on Closed Manifold	p. 203
Operators of Slowly Varying Order	p. 203
Hypoelliptic Operators with Multiple Characteristics	p. 213
Weighted Sobolev Spaces and Hypoelliptic Operators with Multiple Characteristics as Fredholm Operators	p. 225
Interior Boundary Value Problem	p. 239
Algebra of Boundary Value Problems for Class of Pseudodifferential Operators which Change Order on the Boundary	p. 245
Symbols on [actual symbol not reproducible]	p. 245
Classes of operators on Half-Space	p. 251
Weighted Sobolev Spaces	p. 262
Operators on Closed Manifolds with Boundaries	p. 270
An Index Theorem	p. 275
General Schemes of Investigation of Spectral Asymptotics for Degenerate Elliptic Equations	p. 279
General Theorems on Spectral Asymptotics	p. 279
General Schemes of Investigation of Spectral Asymptotics and Generalizations of the Weyl Formula for Degenerate Elliptic Operators	p. 291
Spectral Asymptotics of Degenerate Elliptic Operators	p. 301
Formal Computations of Spectral Asymptotics for Operators of All Types	p. 301
Proof of the Asymptotic Formulae	p. 317
Spectral Asymptotics of Hypoelliptic Operators with Multiple Characteristics	p. 335
Formal Computations of Spectral Asymptotics	p. 335
Proofs of the Asymptotic Formulae	p. 355
A Brief Review of the Bibliography	p. 389
Bibliography	p. 399
Index of Notation	p. 423
Table of Contents provided by Blackwell. All Rights Reserved.

Degenerate Elliptic Equations

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