We study the eigenvalues of the Laplacian with a strong attractive Robin boundary condition in curvilinear polygons. It was known from previous works that the asymptotics of several first eigenvalues is essentially determined by the corner openings, while only rough estimates were available for the next eigenvalues. Under some geometric assumptions, we go beyond the critical eigenvalue number and give a precise asymptotics of any individual eigenvalue by establishing a link with an effective Schrödinger-type operator on the boundary of the domain with boundary conditions at the corners.
Nous étudions les valeurs propres du laplacien avec une condition de Robin fortement attractive dans des polygones curvilignes. Grâce à de précédents travaux, on sait que le comportement asymptotique de quelques premières valeurs propres est essentiellement déterminé par les ouvertures des coins, alors que seules quelques estimées grossières sont disponibles pour les valeurs propres suivantes. Sous certaines hypothèses géométriques, nous allons au-delà du nombre critique de valeurs propres et nous donnons un développement asymptotique précis pour chaque valeur propre individuelle en établissant un lien avec un opérateur effectif de type Schrödinger agissant sur le bord du domaine et muni de conditions aux limites aux coins.
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Published online:
Keywords: Eigenvalue, Laplacian, Robin boundary condition, effective operator, non-smooth domain
Mot clés : Valeur propre, Laplacien, condition aux limites de Robin, opérateur effectif, domaine non-lisse
Khalile, Magda 1; Ourmières-Bonafos, Thomas 2; Pankrashkin, Konstantin 3
@article{AIF_2020__70_5_2215_0, author = {Khalile, Magda and Ourmi\`eres-Bonafos, Thomas and Pankrashkin, Konstantin}, title = {Effective operators for {Robin} eigenvalues in domains with corners}, journal = {Annales de l'Institut Fourier}, pages = {2215--2301}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {70}, number = {5}, year = {2020}, doi = {10.5802/aif.3400}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3400/} }
TY - JOUR AU - Khalile, Magda AU - Ourmières-Bonafos, Thomas AU - Pankrashkin, Konstantin TI - Effective operators for Robin eigenvalues in domains with corners JO - Annales de l'Institut Fourier PY - 2020 SP - 2215 EP - 2301 VL - 70 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3400/ DO - 10.5802/aif.3400 LA - en ID - AIF_2020__70_5_2215_0 ER -
%0 Journal Article %A Khalile, Magda %A Ourmières-Bonafos, Thomas %A Pankrashkin, Konstantin %T Effective operators for Robin eigenvalues in domains with corners %J Annales de l'Institut Fourier %D 2020 %P 2215-2301 %V 70 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3400/ %R 10.5802/aif.3400 %G en %F AIF_2020__70_5_2215_0
Khalile, Magda; Ourmières-Bonafos, Thomas; Pankrashkin, Konstantin. Effective operators for Robin eigenvalues in domains with corners. Annales de l'Institut Fourier, Volume 70 (2020) no. 5, pp. 2215-2301. doi : 10.5802/aif.3400. https://aif.centre-mersenne.org/articles/10.5802/aif.3400/
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