Comptes Rendus
no. 4
Partial differential equations
A note on Sylvester's proof of discreteness of interior transmission eigenvalues
[Une remarque sur la preuve de la distribution discrète des valeurs propres intérieures de transmission de Sylvester]

Présenté par : 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.

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.

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.

Reçu le :
Accepté le :
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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/

[1] A.-S. Bonnet-Ben Dhia; L. Chesnel; H. Haddar On the use of T-coercivity to study the interior transmission eigenvalue problem, C. R. Acad. Sci. Paris, Ser. I, Volume 340 (2011), pp. 647-651

[2] F. Cakoni; H. Haddar (MSRI Publications), Volume vol. 60 (2012), pp. 527-578

[3] D. Colton; A. Kirsch; L. Päivärinta Far field patterns for acoustic waves in an inhomogeneous medium, SIAM J. Math. Anal., Volume 20 (1989), pp. 1472-1483

[4] D.L. Colton; R. Kress Inverse Acoustic and Electromagnetic Scattering Theory, Springer, 2013

[5] D. Colton; L. Päivärinta; J. Sylvester The interior transmission problem, Inverse Probl. Imaging, Volume 1 (2007), pp. 13-28

[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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