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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

6 Issues per year

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A variational approach to the Cauchy problem for nonlinear elliptic differential equations

I. Ly1 / N. Tarkhanov2

1Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany. Email:

2Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany. Email:

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 17, Issue 6, Pages 595–610, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: 10.1515/JIIP.2009.037, August 2009

Publication History

Received:
2009-01-27
Published Online:
2009-08-19

Abstract

We discuss the relaxation of a class of nonlinear elliptic Cauchy problems with data on a piece S of the boundary surface by means of a variational approach known in the optimal control literature as “equation error method”. By the Cauchy problem is meant any boundary value problem for an unknown function y in a domain χ with the property that the data on S, if combined with the differential equations in χ, allow one to determine all derivatives of y on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy–Kovalevskaya theorem. We also admit overdetermined elliptic systems, in which case the set of those Cauchy data on S for which the Cauchy problem is solvable is very “thin”. For this reason we discuss a variational setting of the Cauchy problem which always possesses a generalised solution.

Key words.: Nonlinear PDE; Cauchy problem; Euler's equations; inverse problem

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