Strongly oscillating singularities for the interior transmission eigenvalue problem

Anne-Sophie Bonnet-Ben Dhia 1 Lucas Chesnel 1
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : In this paper, we investigate a two-dimensional interior transmission eigenvalue problem for an inclusion made of a composite material. We consider configurations where the difference between the parameters of the composite material and those of the background changes sign on the boundary of the inclusion. In a first step, under some assumptions on the parameters, we extend the variational approach of the T-coercivity to prove that the transmission eigenvalues form at most a discrete set. In the process, we also provide localization results. Then, we study what happens when these assumptions are not satisfied. The main idea is that, due to very strong singularities that can occur at the boundary, the problem may lose Fredholmness in the natural H1 framework. Using Kondratiev theory, we propose a new functional framework where the Fredholm property is restored.
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https://hal.inria.fr/hal-00937690
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Submitted on : Tuesday, January 28, 2014 - 5:06:44 PM
Last modification on : Wednesday, July 3, 2019 - 10:48:03 AM

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Anne-Sophie Bonnet-Ben Dhia, Lucas Chesnel. Strongly oscillating singularities for the interior transmission eigenvalue problem. Inverse Problems, IOP Publishing, 2013, 19(10), pp.104004. ⟨10.1088/0266-5611/29/10/104004⟩. ⟨hal-00937690⟩

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