Comptes Rendus
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.

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.

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.

Received:
Accepted:
Published online:
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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Cited by 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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