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The Analysis of Solutions of Elliptic Equations

  • Nikolai N. Tarkhanov

Part of the Mathematics and Its Applications book series (MAIA, volume 406)

Table of contents

  1. Front Matter
    Pages i-xx
  2. Nikolai N. Tarkhanov
    Pages 1-3
  3. Nikolai N. Tarkhanov
    Pages 5-66
  4. Nikolai N. Tarkhanov
    Pages 67-114
  5. Nikolai N. Tarkhanov
    Pages 191-270
  6. Nikolai N. Tarkhanov
    Pages 271-318
  7. Nikolai N. Tarkhanov
    Pages 319-344
  8. Nikolai N. Tarkhanov
    Pages 345-371
  9. Nikolai N. Tarkhanov
    Pages 373-409
  10. Nikolai N. Tarkhanov
    Pages 411-449
  11. Back Matter
    Pages 451-484

About this book

Introduction

This book is intended as a continuation of my book "Parametrix Method in the Theory of Differential Complexes" (see [291]). There, we considered complexes of differential operators between sections of vector bundles and we strived more than for details. Although there are many applications to for maximal generality overdetermined systems, such an approach left me with a certain feeling of dissat- faction, especially since a large number of interesting consequences can be obtained without a great effort. The present book is conceived as an attempt to shed some light on these new applications. We consider, as a rule, differential operators having a simple structure on open subsets of Rn. Currently, this area is not being investigated very actively, possibly because it is already very highly developed actively (cf. for example the book of Palamodov [213]). However, even in this (well studied) situation the general ideas from [291] allow us to obtain new results in the qualitative theory of differential equations and frequently in definitive form. The greater part of the material presented is related to applications of the L- rent series for a solution of a system of differential equations, which is a convenient way of writing the Green formula. The culminating application is an analog of the theorem of Vitushkin [303] for uniform and mean approximation by solutions of an elliptic system. Somewhat afield are several questions on ill-posedness, but the parametrix method enables us to obtain here a series of hitherto unknown facts.

Keywords

Cauchy problem Potential theory differential equation differential operator functional analysis partial differential equation

Authors and affiliations

  • Nikolai N. Tarkhanov
    • 1
  1. 1.Institute of PhysicsKrasnoyarskRussia

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-015-8804-1
  • Copyright Information Springer Science+Business Media B.V. 1997
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-4845-5
  • Online ISBN 978-94-015-8804-1
  • Buy this book on publisher's site