Synopses & Reviews
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.
Review
`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
Review
`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
Table of Contents
Prefaces. List of Main Notations.
1. Removable Singularities.
2. Laurent Series.
3. Representation of Solutions with Non-Discrete Singularities.
4. Uniform Approximation.
5. Mean Approximation.
6. BMO Approximation.
7. Conditional Stability.
8. The Cauchy Problem.
9. Quasiconformality. Bibliography. Name Index. Subject Index. Index of Notation.