Comptes Rendus
New asymptotic effects for the spectrum of problems on concentrated masses near the boundary
[Nouveaux effets asymptotiques pour le spectre des problèmes avec des masses concentrées près de la frontière]
Comptes Rendus. Mécanique, Volume 337 (2009) no. 8, pp. 585-590.

On considére des problèmes spectraux pour l'opérateur de Laplace dans un domaine bornée ΩR2 avec des conditions de Dirichlet et Neumann respectivement sur la frontière. On suppose que la frontière ∂Ω est régulière par morceaux tandis que la fonction densité prend la valeur 1+εmχε dans Ω, oú ε>0 est un petit paramètre, mR, et χε est la fonction caractéristique de l'union des petites ensembles ωε0ωεJ1 (les masses concentrés), qui sont répartis périodiquement prés d'un segment droite Γ de la frontière, ΓΩ. Nous décrivons le comportement asymptotique des valeurs propres de ces deux problèmes lorsque ε0.

The Dirichlet and Neumann spectral problems for the Laplace operator in a bounded domain ΩR2 are considered. We assume that Ω has a piecewise smooth boundary ∂Ω and the density function is equal to 1+εmχε in Ω, where ε>0 is a small parameter, mR and χε is the characteristic function of the union ωε0ωεJ1 of small sets (the concentrated masses) distributed periodically near a straight segment ΓΩ. We describe asymptotics for the eigenelements of both problems as ε0.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2009.07.002
Keywords: Boundary homogenization, Spectral analysis, Concentrated masses, Asymptotic expansions
Mot clés : Homogénéisation des frontières, Analyse spectrale, Masses concentrées, Développements asymptotiques

Sergey A. Nazarov 1 ; Eugenia Pérez 2

1 Institute of Mechanical Engineering Problems, V.O., Bol'shoi pr., 61, 199178, St.-Petersburg, Russia
2 Departamento de Matemática Aplicada y Ciencias de la Computación, Universidad de Cantabria, Avenida de las Castros s/n, 39005 Santander, Spain
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Sergey A. Nazarov; Eugenia Pérez. New asymptotic effects for the spectrum of problems on concentrated masses near the boundary. Comptes Rendus. Mécanique, Volume 337 (2009) no. 8, pp. 585-590. doi : 10.1016/j.crme.2009.07.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2009.07.002/

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Cité par Sources :

The first author acknowledges the support by RFFI, grant 09-01-00759. The second author acknowledges the support by the Spanish MEC, MTM2005-07720. The work has also been partially supported by the MEC, SAB2005-0175.

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