Transmission eigenvalues for  inhomogeneous media containing obstacles

Fioralba Cakoni; Anne Cossonnière; Houssem Haddar

doi:10.3934/ipi.2012.6.373

Article Contents

2012, Volume 6, Issue 3: 373-398. Doi: 10.3934/ipi.2012.6.373

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Transmission eigenvalues for inhomogeneous media containing obstacles

1.
Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716-2553
2.
CERFACS, 42 avenue Gaspard Coriolis, 31057 Toulouse Cedex 01
3.
INRIA Saclay Ile de France/CMAP Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex

Received: November 2011

Revised: June 2012

Published online: September 2012

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Abstract

We consider the interior transmission problem corresponding to the inverse scattering by an inhomogeneous (possibly anisotropic) media in which an impenetrable obstacle with Dirichlet boundary conditions is embedded. Our main focus is to understand the associated eigenvalue problem, more specifically to prove that the transmission eigenvalues form a discrete set and show that they exist. The presence of Dirichlet obstacle brings new difficulties to already complicated situation dealing with a non-selfadjoint eigenvalue problem. In this paper, we employ a variety of variational techniques under various assumptions on the index of refraction as well as the size of the Dirichlet obstacle.

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Mathematics Subject Classification: Primary: 35R30, 35Q60, 35J40, 78A25.

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