The Analysis of Solutions of Elliptic Equations

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.

Klappentext

This volume focuses on the analysis of solutions to general elliptic equations. A wide range of topics is touched upon, such as removable singularities, Laurent expansions, approximation by solutions, Carleman formulas, quasiconformality. While the basic setting is the Dirichlet problem for the Laplacian, there is some discussion of the Cauchy problem. Care is taken to distinguish between results which hold in a very general setting (arbitrary elliptic equation with the unique continuation property) and those which hold under more restrictive assumptions on the differential operators (homogeneous, of first order). Some parallels to the theory of functions of several complex variables are also sketched. Audience: This book will be of use to postgraduate students and researchers whose work involves partial differential equations, approximations and expansion, several complex variables and analytic spaces, potential theory and functional analysis. It can be recommended as a text for seminars and courses, as well as for independent study.

Zusammenfassung
`Summarising this is a well-written book. It provides the complete survey of the history and the recent state of the subject topic. The book will be of use to postgraduate students and researchers in PDE's, in theory of approximations and expansions, in potential theory or in functional analysis. It is warmly recommended as a text for seminars and courses, as well as for independent study for everyone with a basic knowledge on distributions and pseudodifferential operators.'
Acta Scientiarum Mathematicarum

Inhalt
List of Main Notations.- 1 Removable Singularities.- 2 Laurent Series.- 3 Representation of Solutions with Non-Discrete Singularities.- 4 Uniform Approximation.- 6 BMO Approximation.- 7 Conditional Stability.- 8 The Cauchy Problem.- 9 Quasiconformality.- Name Index.- Index of Notation.