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

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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/

Bibliographie
Cité par

[1] M. Lobo; E. Pérez Math. Models Methods Appl. Sci., 5 (1995) no. 5, pp. 565-585

[2] Y. Golovaty; D. Gómez; M. Lobo; E. Pérez Math. Models Methods Appl. Sci., 14 (2004), pp. 987-1034

[3] D. Gómez; M. Lobo; S.A. Nazarov; E. Pérez J. Math. Pures Appl., 85 (2006), pp. 598-632

[4] D. Gómez; M. Lobo; S.A. Nazarov; E. Pérez J. Math. Pures Appl., 86 (2006), pp. 369-402

[5] O.A. Oleinik; A.S. Shamaev; G.A. Yosifian Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992

[6] M. Lobo; E. Pérez C. R. Mecanique, 331 (2003), pp. 303-317

[7] E. Pérez Multi Scale Problems and Asymptotic Analysis, GAKUTO Internat. Ser. Math. Sci. Appl., vol. 24, Gakkotosho, Tokyo, 2006, pp. 311-323

[8] E. Pérez Discrete Cont. Dyn. Syst. Ser. B, 7 (2007) no. 4, pp. 859-883

[9] S.A. Nazarov, E. Pérez, Higher order terms of asymptotics for eigenelements in vibrating systems with many concentrated masses, in preparation

[10] M. Maz'ya; S.A. Nazarov; B.A. Plamenevskij Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, vols. I and II, Birkhäuser Verlag, Basel, 2000

[11] S.A. Nazarov; B.A. Plamenevskii Leningrad Math. J., 2 (1991), pp. 287-311

[12] G. Nguetseng; E. Sanchez-Palencia Local Effects in the Analysis of Structures, Stud. Appl. Mech., vol. 12, Elsevier, Amsterdam, 1985, pp. 55-74

[13] M. Lobo; E. Pérez Math. Models Methods Appl. Sci., 3 (1993) no. 2, pp. 249-273

[14] M. Lobo; E. Pérez Math. Methods Appl. Sci., 24 (2001) no. 1, pp. 59-80

[15] G.A. Chechkin C. R. Mecanique, 332 (2004), pp. 949-954

[16] G.A. Chechkin; E. Pérez; E.I. Yablokova Indiana Univ. Math. J., 54 (2005), pp. 321-348

[17] T. Mel'nyk Math. Models Methods Appl. Sci., 11 (2001) no. 6, pp. 1001-1027

[18] D. Leguillon; E. Sanchez-Palencia Computation of Singular Solutions in Elliptic Problems and Elasticity, Masson, Paris, 1987

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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