The Cauchy Problem for Solutions of Elliptic Equations

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9783527400584: The Cauchy Problem for Solutions of Elliptic Equations
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The book is an attempt to bring together various topics in partial differential equations related to the Cauchy problem for solutions of an elliptic equation. Ever since Hadamard, the Cauchy problem for solutions of elliptic equations has been known to be ill-posed. It is conditionally stable, just as is the case for even the simplest problems of analytic continuation of functions given on a subset of the boundary. (Such problems of analytic continuation served as a paradigm for the treatment here.) The study of the Cauchy problem is carried out in three directions: determining the degree of instability, which is connected with sharp theorems on approximation by solutions of an elliptic equation; finding solvability conditions, which is based on the development of Hilbert space methods in the Cauchy problem; and reconstructing solutions via their Cauchy data, which requires efficient ways of approximation. A wide range of topics is touched upon, among them are function spaces on compact sets, boundedness theorems for pseudodifferential operators in nonlocal spaces, nonlinear capacity and removable singularities, fundamental solutions, capacitary criteria for approximation by solutions of elliptic equations, and weak boundary values of the solutions. The theory applies as well to the Cauchy problem for solution of
overdetermined elliptic systems.

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From the Publisher:

This book examines the Cauchy problem for elliptic equations under the aspects of approximation, solvability, and reconstruction of solutions via their initial data. It also explores the problem of analytic continuation of functions from a boundary subset.

About the Author:

Author's affiliation:
Nikolai N. Tarkhanov, Professor, University of Krasnoyarsk, University of Potsdam, Max Planck Research Group "Partial Differential Equations and Complex Analysis"

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Other Popular Editions of the Same Title

9783055016639: The Cauchy Problem for Solutions of Elliptic Equations (Mathematical Topics)

Featured Edition

ISBN 10:  3055016637 ISBN 13:  9783055016639
Publisher: Vch Pub, 1995
Hardcover