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.
Accepté le :
Publié le :
@article{CRMATH_2016__354_4_377_0, author = {Kirsch, Andreas}, title = {A note on {Sylvester's} proof of discreteness of interior transmission eigenvalues}, journal = {Comptes Rendus. Math\'ematique}, pages = {377--382}, publisher = {Elsevier}, volume = {354}, number = {4}, year = {2016}, doi = {10.1016/j.crma.2016.01.015}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2016.01.015/} }
TY - JOUR AU - Kirsch, Andreas TI - A note on Sylvester's proof of discreteness of interior transmission eigenvalues JO - Comptes Rendus. Mathématique PY - 2016 SP - 377 EP - 382 VL - 354 IS - 4 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2016.01.015/ DO - 10.1016/j.crma.2016.01.015 LA - en ID - CRMATH_2016__354_4_377_0 ER -
%0 Journal Article %A Kirsch, Andreas %T A note on Sylvester's proof of discreteness of interior transmission eigenvalues %J Comptes Rendus. Mathématique %D 2016 %P 377-382 %V 354 %N 4 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2016.01.015/ %R 10.1016/j.crma.2016.01.015 %G en %F CRMATH_2016__354_4_377_0
Kirsch, Andreas. A note on Sylvester's proof of discreteness of interior transmission eigenvalues. Comptes Rendus. Mathématique, Tome 354 (2016) no. 4, pp. 377-382. doi : 10.1016/j.crma.2016.01.015. http://archive.numdam.org/articles/10.1016/j.crma.2016.01.015/
[1] 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] (MSRI Publications), Volume vol. 60 (2012), pp. 527-578
[3] Far field patterns for acoustic waves in an inhomogeneous medium, SIAM J. Math. Anal., Volume 20 (1989), pp. 1472-1483
[4] Inverse Acoustic and Electromagnetic Scattering Theory, Springer, 2013
[5] The interior transmission problem, Inverse Probl. Imaging, Volume 1 (2007), pp. 13-28
[6] Elliptic in the interior transmission problem in anisotropic media, SIAM J. Math. Anal., Volume 44 (2012), pp. 1165-1174
[7] Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem, Inverse Probl., Volume 29 (2013), p. 104003
[8] Finite Element Methods for Maxwell's Equations, Oxford University Press, 2003
[9] Transmission eigenvalues, SIAM J. Math. Anal., Volume 40 (2008), pp. 738-753
[10] Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., Volume 44 (2011) no. 1, pp. 341-354
Cité par Sources :