Comptes Rendus
no. 4
Partial differential equations
A note on Sylvester's proof of discreteness of interior transmission eigenvalues

Presented by: Haïm Brézis

Andreas Kirsch1
1 Karlsruhe Institute of Technology (KIT), Department of Mathematics, 76131 Karlsruhe, Germany
Comptes Rendus. Mathématique, Volume 354 (2016) no. 4, pp. 377-382.

It has been shown by Sylvester (2011) [10] that the set of interior transmission eigenvalues forms a discrete set if the contrast does not change its sign in a neighborhood of the boundary. In this short note, we give a more elementary proof of this fact using the classical inf–sup conditions of Babuška–Brezzi.

Il a été démontré par Sylvester (2011) [10] que l'ensemble des valeurs propres intérieures de transmission constitue un ensemble discret si le contraste ne change pas de signe dans un voisinage du bord. Nous donnons une preuve plus élémentaire de ce fait en utilisant les conditions classiques « inf–sup » de Babuška–Brezzi.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.01.015

Andreas Kirsch 1

1 Karlsruhe Institute of Technology (KIT), Department of Mathematics, 76131 Karlsruhe, Germany 
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Andreas Kirsch. A note on Sylvester's proof of discreteness of interior transmission eigenvalues. Comptes Rendus. Mathématique, Volume 354 (2016) no. 4, pp. 377-382. doi : 10.1016/j.crma.2016.01.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.01.015/

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[6] E. Lakshtanov; B. Vainberg Elliptic in the interior transmission problem in anisotropic media, SIAM J. Math. Anal., Volume 44 (2012), pp. 1165-1174

[7] E. Lakshtanov; B. Vainberg Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem, Inverse Probl., Volume 29 (2013), p. 104003

[8] P. Monk Finite Element Methods for Maxwell's Equations, Oxford University Press, 2003

[9] L. Päivärinta; J. Sylvester Transmission eigenvalues, SIAM J. Math. Anal., Volume 40 (2008), pp. 738-753

[10] J. Sylvester Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., Volume 44 (2011) no. 1, pp. 341-354

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