Comptes Rendus
no. 11-12
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 Dhia1 ; Lucas Chesnel1 ; Houssem Haddar2
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.

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 :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.05.008
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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     title = {On the use of {\protect\emph{T}-coercivity} to study the interior transmission eigenvalue problem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {647--651},
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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/

[1] A.-S. Bonnet-Ben Dhia; L. Chesnel; P. Ciarlet Optimality of T-coercivity for scalar interface problems between dielectrics and metamaterials, 2010 | HAL

[2] A.-S. Bonnet-Ben Dhia; P. Ciarlet; C.M. Zwölf Time harmonic wave diffraction problems in materials with sign-shifting coefficients, J. Comput. Appl. Math., Volume 234 (2010), pp. 1912-1919 (Corrigendum: J. Comput. Appl. Math., 234, 2010, pp. 2616)

[3] F. Cakoni; D. Colton; H. Haddar The linear sampling method for anisotropic media, J. Comput. Appl. Math., Volume 146 (2002), pp. 285-299

[4] F. Cakoni; D. Gintides; H. Haddar The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., Volume 42 (2010), pp. 237-255

[5] F. Cakoni; A. Kirsch On the interior transmission eigenvalue problem, Int. J. Comp. Sci. Math., Volume 3 (2010) no. 1–2, pp. 142-167

[6] D. Colton; L. Päivärinta; J. Sylvester The interior transmission problem, Inverse Problems and Imaging, Volume 1 (2007) no. 1, pp. 13-28

[7] M. Hitrik; K. Krupchyk; P. Ola; L. Païvärinta The interior transmission problem and bounds on transmission eigenvalues, 2010 (preprint) | arXiv

[8] B.P. Rynne; B.D. Sleeman The interior transmission problem and inverse scattering from inhomogeneous media, SIAM J. Math. Anal., Volume 22 (1991), pp. 1755-1762

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