New asymptotic effects for the spectrum of problems on concentrated masses near the boundary

Sergey A. Nazarov; Eugenia Pérez

doi:10.1016/j.crme.2009.07.002

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]

Présenté par : Evariste Sanchez-Palencia

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

Comptes Rendus. Mécanique, Volume 337 (2009) no. 8, pp. 585-590.

Résumé
Abstract

On considére des problèmes spectraux pour l'opérateur de Laplace dans un domaine bornée $Ω \subset R^{2}$ 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 $1 + ε^{- m} χ_{ε}$ dans Ω, oú $ε > 0$ est un petit paramètre, $m \in R$ , et $χ_{ε}$ est la fonction caractéristique de l'union des petites ensembles $ω_{ε}^{0} \cup \dots \cup ω_{ε}^{J - 1}$ (les masses concentrés), qui sont répartis périodiquement prés d'un segment droite Γ de la frontière, $Γ \subset \partial Ω$ . Nous décrivons le comportement asymptotique des valeurs propres de ces deux problèmes lorsque $ε \to 0$ .

The Dirichlet and Neumann spectral problems for the Laplace operator in a bounded domain $Ω \subset R^{2}$ are considered. We assume that Ω has a piecewise smooth boundary ∂Ω and the density function is equal to $1 + ε^{- m} χ_{ε}$ in Ω, where $ε > 0$ is a small parameter, $m \in R$ and $χ_{ε}$ is the characteristic function of the union $ω_{ε}^{0} \cup \dots \cup ω_{ε}^{J - 1}$ of small sets (the concentrated masses) distributed periodically near a straight segment $Γ \subset \partial Ω$ . We describe asymptotics for the eigenelements of both problems as $ε \to 0$ .

Reçu le : 2009-06-26
Accepté le : 2009-07-16
Publié le : 2009-08-01

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

Affiliations des auteurs :

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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     author = {Sergey A. Nazarov and Eugenia P\'erez},
     title = {New asymptotic effects for the spectrum of problems on concentrated masses near the boundary},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {585--590},
     publisher = {Elsevier},
     volume = {337},
     number = {8},
     year = {2009},
     doi = {10.1016/j.crme.2009.07.002},
     language = {en},
}

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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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