[Utilisation de la T-coercivité pour lʼétude du problème de transmission intérieur]
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.
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@article{CRMATH_2011__349_11-12_647_0,
author = {Bonnet-Ben Dhia, Anne-Sophie and Chesnel, Lucas and Haddar, Houssem},
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},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.05.008/}
}
TY - JOUR AU - Bonnet-Ben Dhia, Anne-Sophie AU - Chesnel, Lucas AU - Haddar, Houssem TI - On the use of T-coercivity to study the interior transmission eigenvalue problem JO - Comptes Rendus. Mathématique PY - 2011 SP - 647 EP - 651 VL - 349 IS - 11-12 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.05.008/ DO - 10.1016/j.crma.2011.05.008 LA - en ID - CRMATH_2011__349_11-12_647_0 ER -
%0 Journal Article %A Bonnet-Ben Dhia, Anne-Sophie %A Chesnel, Lucas %A Haddar, Houssem %T On the use of T-coercivity to study the interior transmission eigenvalue problem %J Comptes Rendus. Mathématique %D 2011 %P 647-651 %V 349 %N 11-12 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.05.008/ %R 10.1016/j.crma.2011.05.008 %G en %F CRMATH_2011__349_11-12_647_0
Bonnet-Ben Dhia, Anne-Sophie; Chesnel, Lucas; Haddar, Houssem. On the use of T-coercivity to study the interior transmission eigenvalue problem. Comptes Rendus. Mathématique, Tome 349 (2011) no. 11-12, pp. 647-651. doi : 10.1016/j.crma.2011.05.008. http://www.numdam.org/articles/10.1016/j.crma.2011.05.008/
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