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Comptes Rendus Mathématique
Volume 351, n° 13-14
pages 517-521 (juillet 2013)
Doi : 10.1016/j.crma.2013.07.022
Received : 19 June 2013 ;  accepted : 31 July 2013
On the asymptotics of a Robin eigenvalue problem
Asymptotique dʼun problème aux valeurs propres avec condition de Robin
 

Fioralba Cakoni a , Nicolas Chaulet b , Houssem Haddar c
a Department of Mathematical Sciences, University of Delaware, USA 
b Department of Mathematics, University College London, UK 
c CMAP, École polytechnique, Palaiseau, France 

Abstract

The considered Robin problem can formally be seen as a small perturbation of a Dirichlet problem. However, due to the sign of the impedance value, its associated eigenvalues converge point-wise to −∞ as the perturbation goes to zero. We prove in this case that Dirichlet eigenpairs are the only accumulation points of the Robin eigenpairs with normalized eigenvectors. We then provide a criterion to select accumulating sequences of eigenvalues and eigenvectors and exhibit their full asymptotic with respect to the small parameter.

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Résumé

Le problème de Robin que lʼon considère peut formellement être vu comme une petite perturbation dʼun problème de Dirichlet. Néanmoins, à cause du signe de lʼimpédance, ses valeurs propres vont ponctuellement vers −∞ lorsque le petit paramètre tend vers 0. Nous montrons néanmoins que les couples valeurs–vecteurs propres du problème de Dirichlet sont les seuls points dʼaccumulation des couples valeurs–vecteurs propres de Robin associés à des suites de vecteurs propres normalisés. Nous proposons un critère qui permet de sélectionner les suites de valeurs propres et de vecteurs propres qui sʼaccumulent sur les valeurs propres et les vecteurs propres de Dirichlet, et nous donnons et justifions leur développement asymptotique complet par rapport au petit paramètre.

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