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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Volume 17, Issue 7 (Jan 2009)


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

W. Arendt / T. Regińska
Published Online: 2009-10-12 | DOI: https://doi.org/10.1515/JIIP.2009.041


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

About the article

Received: 2008-03-15

Published Online: 2009-10-12

Published in Print: 2009-10-01

Citation Information: Journal of Inverse and Ill-posed Problems, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/JIIP.2009.041.

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