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.

Abstract (VO)
Résumé

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 $ε \to 0$ .

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 $ε \to 0$ .

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
Mots-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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     title = {New asymptotic effects for the spectrum of problems on concentrated masses near the boundary},
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     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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Yuriy Golovaty Membranes with thin and heavy inclusions: Asymptotics of spectra, Asymptotic Analysis, Volume 130 (2022) no. 1-2, p. 23 | DOI:10.3233/asy-211743
A. G. Chechkina On the Behavior of the Spectrum of a Perturbed Steklov Boundary Value Problem with a Weak Singularity, Differential Equations, Volume 57 (2021) no. 10, p. 1382 | DOI:10.1134/s0012266121100128
Fedor L. Bakharev; M. Eugenia Pérez Spectral gaps for the Dirichlet‐Laplacian in a 3‐D waveguide periodically perturbed by a family of concentrated masses, Mathematische Nachrichten, Volume 291 (2018) no. 4, p. 556 | DOI:10.1002/mana.201600270
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D. Gómez; S. A. Nazarov; M. E. Pérez Homogenization of Winkler–Steklov spectral conditions in three-dimensional linear elasticity, Zeitschrift für angewandte Mathematik und Physik, Volume 69 (2018) no. 2 | DOI:10.1007/s00033-018-0927-8
Giuseppe Cardone; Andrii Khrabustovskyi Neumann spectral problem in a domain with very corrugated boundary, Journal of Differential Equations, Volume 259 (2015) no. 6, p. 2333 | DOI:10.1016/j.jde.2015.03.031
G. A. Chechkin; T. A. Mel'nyk Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses, Mathematical Methods in the Applied Sciences, Volume 37 (2014) no. 1, p. 56 | DOI:10.1002/mma.2785
Miguel Lobo; M. Eugenia PÃ©rez Long time approximations for solutions of wave equations associated with the Steklov spectral homogenization problems, Mathematical Methods in the Applied Sciences (2009), p. n/a | DOI:10.1002/mma.1256

Cité par 9 documents. Sources : Crossref

^☆ 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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