Asymptotics of the solution of a Dirichlet spectral problem in a junction with highly oscillating boundary

Youcef Amirat; Gregory A. Chechkin; Rustem R. Gadyl'shin

doi:10.1016/j.crme.2008.06.008

Asymptotics of the solution of a Dirichlet spectral problem in a junction with highly oscillating boundary

Presented by: Évariste Sanchez-Palencia

Youcef Amirat ¹; Gregory A. Chechkin ^2,³; Rustem R. Gadyl'shin ⁴

¹ Laboratoire de mathématiques, CNRS UMR 6620, Université Blaise-Pascal, 63177 Aubière cedex, France
² Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow Lomonosov State University, Moscow 119991, Russia
³ Narvik University College, Postboks 385, 8505 Narvik, Norway
⁴ Department of Mathematical Analysis, Faculty of Physics and Mathematics, Bashkir State Pedagogical University, Ufa 450000, Russia

Comptes Rendus. Mécanique, Volume 336 (2008) no. 9, pp. 693-698.

Abstract
Résumé

We study the asymptotic behavior of the eigenelements of the Dirichlet problem for the Laplacian in a bounded domain, a part of whose boundary, depending on a small parameter ε, is highly oscillating; the frequency of oscillations of the boundary is of order ε and the amplitude is fixed. We present second-order asymptotic approximations, as $ε \to 0$ , of the eigenelements in the case of simple eigenvalues of the limit problem.

Nous étudions le comportement asymptotique des éléments propres du problème de Dirichlet pour le Laplacien dans un domaine borné dont une partie de la frontière, dépendant d'un petit paramètre ε, est fortement oscillante ; la fréquence des oscillations est d'ordre ε et leur amplitude est fixe. Nous présentons des approximations asymptotiques d'ordre deux des éléments propres dans le cas de valeurs propres simples du problème limite.

Received: 2008-06-14
Accepted: 2008-06-19
Published online: 2008-08-01

DOI: 10.1016/j.crme.2008.06.008

Keywords: Asymptotic expansion, Spectral problem, Oscillating boundary
Mots-clés : Développement asymptotique, Problème spectral, Frontière oscillante

Author's affiliations:

Youcef Amirat ¹; Gregory A. Chechkin ^{2, 3}; Rustem R. Gadyl'shin ⁴

¹ Laboratoire de mathématiques, CNRS UMR 6620, Université Blaise-Pascal, 63177 Aubière cedex, France
² Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow Lomonosov State University, Moscow 119991, Russia
³ Narvik University College, Postboks 385, 8505 Narvik, Norway
⁴ Department of Mathematical Analysis, Faculty of Physics and Mathematics, Bashkir State Pedagogical University, Ufa 450000, Russia

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     author = {Youcef Amirat and Gregory A. Chechkin and Rustem R. Gadyl'shin},
     title = {Asymptotics of the solution of a {Dirichlet} spectral problem in a junction with highly oscillating boundary},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {693--698},
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     year = {2008},
     doi = {10.1016/j.crme.2008.06.008},
     language = {en},
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Youcef Amirat; Gregory A. Chechkin; Rustem R. Gadyl'shin. Asymptotics of the solution of a Dirichlet spectral problem in a junction with highly oscillating boundary. Comptes Rendus. Mécanique, Volume 336 (2008) no. 9, pp. 693-698. doi : 10.1016/j.crme.2008.06.008. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2008.06.008/

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