On the use of <i>T</i>-coercivity to study the interior transmission eigenvalue problem

Anne-Sophie Bonnet-Ben Dhia; Lucas Chesnel; Houssem Haddar

doi:10.1016/j.crma.2011.05.008

Partial Differential Equations

On the use of T-coercivity to study the interior transmission eigenvalue problem
[Utilisation de la T-coercivité pour lʼétude du problème de transmission intérieur]

Présenté par : Evariste Sanchez-Palencia

Anne-Sophie Bonnet-Ben Dhia ¹ ; Lucas Chesnel ¹ ; Houssem Haddar ²

¹ POEMS, UMR 7231 CNRS-INRIA-ENSTA, 32, boulevard Victor, 75739 Paris cedex 15, France
² INRIA Saclay Ile de France / École polytechnique, route de Saclay, 91128 Palaiseau cedex, France

Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 647-651.

Résumé
Abstract

Dans cette Note, nous utilisons la méthode de la T-coercivité pour étudier le problème de transmission intérieur. En particulier, nous prouvons que ce problème, qui apparaît lorsquʼon cherche à reconstruire le support dʼune inclusion dans un milieu homogène, est de type Fredholm et que lʼensemble de ses valeurs propres est discret. Notre approche permet de traiter des cas pour lesquels la différence entre lʼindice de lʼinclusion et celui du milieu environnant change de signe, cas qui nʼétaient pas couverts par les techniques classiques de la littérature. Nous donnons également des inégalités de type Faber–Krahn dans ces situations.

In this Note, we investigate the so-called interior transmission problem using the T-coercivity approach. In particular, we prove that this problem, which appears when one is interested in the reconstruction of the support of an inclusion embedded in a homogeneous medium, is of Fredholm type and that so-called transmission eigenvalues form at most a discrete set. Our approach treats cases where the difference between the inclusion index and the background index can change sign, which are not covered by other techniques that can be found in the literature. We also provide Faber–Krahn type inequalities associated with this general case.

Reçu le : 2011-03-18
Accepté le : 2011-05-13
Publié le : 2011-06-01

DOI : 10.1016/j.crma.2011.05.008

Affiliations des auteurs :

Anne-Sophie Bonnet-Ben Dhia ¹ ; Lucas Chesnel ¹ ; Houssem Haddar ²

¹ POEMS, UMR 7231 CNRS-INRIA-ENSTA, 32, boulevard Victor, 75739 Paris cedex 15, France
² INRIA Saclay Ile de France / École polytechnique, route de Saclay, 91128 Palaiseau cedex, France

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     author = {Anne-Sophie Bonnet-Ben Dhia and Lucas Chesnel and Houssem Haddar},
     title = {On the use of {\protect\emph{T}-coercivity} to study the interior transmission eigenvalue problem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {647--651},
     publisher = {Elsevier},
     volume = {349},
     number = {11-12},
     year = {2011},
     doi = {10.1016/j.crma.2011.05.008},
     language = {en},
}

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Anne-Sophie Bonnet-Ben Dhia; Lucas Chesnel; Houssem Haddar. On the use of T-coercivity to study the interior transmission eigenvalue problem. Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 647-651. doi : 10.1016/j.crma.2011.05.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.05.008/

Bibliographie
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