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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


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An ill-posed boundary value problem for the Helmholtz equation on Lipschitz domains

W. Arendt1 / T. Regińska2

1Institute of Applied Analysis, University of Ulm, D-89069 Ulm, Germany. Email:

2Institute of Mathematics, Polish Academy of Sciences, 00-956 Warsaw, Poland. Email:

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 17, Issue 7, Pages 703–711, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: 10.1515/JIIP.2009.041, October 2009

Publication History

Received:
2008-03-15
Published Online:
2009-10-12

Abstract

The paper is concerned with properties of an ill-posed problem for the Helmholtz equation when Dirichlet and Neumann conditions are given only on a part Γ of the boundary ∂Ω. We present an equivalent formulation of this problem in terms of a moment problem defined on the part of the boundary where no boundary conditions are imposed. Using a weak definition of the normal derivative, we prove the equivalence between these two problems for an arbitrary Lipschitz domain in . Moreover, uniqueness of the solution is proved for the general case when Γ is a non-empty open subset of the Lipschitz boundary.

Key words.: Ill-posed problem; Helmholtz equation; Lipschitz domain; weak normal derivative; moment problem; uniqueness of the inverse problem

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