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 (Original language)
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

@article{CRMECA_2008__336_9_693_0,
     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},
     publisher = {Elsevier},
     volume = {336},
     number = {9},
     year = {2008},
     doi = {10.1016/j.crme.2008.06.008},
     language = {en},
}

TY  - JOUR
AU  - Youcef Amirat
AU  - Gregory A. Chechkin
AU  - Rustem R. Gadyl'shin
TI  - Asymptotics of the solution of a Dirichlet spectral problem in a junction with highly oscillating boundary
JO  - Comptes Rendus. Mécanique
PY  - 2008
SP  - 693
EP  - 698
VL  - 336
IS  - 9
PB  - Elsevier
DO  - 10.1016/j.crme.2008.06.008
LA  - en
ID  - CRMECA_2008__336_9_693_0
ER  -

%0 Journal Article
%A Youcef Amirat
%A Gregory A. Chechkin
%A Rustem R. Gadyl'shin
%T Asymptotics of the solution of a Dirichlet spectral problem in a junction with highly oscillating boundary
%J Comptes Rendus. Mécanique
%D 2008
%P 693-698
%V 336
%N 9
%I Elsevier
%R 10.1016/j.crme.2008.06.008
%G en
%F CRMECA_2008__336_9_693_0

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/

References
Cited by

[1] Y. Amirat; G.A. Chechkin; R.R. Gadyl'shin Asymptotics for eigenelements of Laplacian in domain with oscillating boundary: Multiple eigenvalues, Appl. Anal., Volume 86 (2007) no. 7, pp. 873-897

[2] S.A. Nazarov Binomial asymptotic behavior of solutions of spectral problems with singular perturbations, Mat. Sb., Volume 181 (1990) no. 3, pp. 291-320 (Translation in Math. USSR-Sb., 69, 2, 1991, pp. 307-340)

[3] T.A. Mel'nyk On free vibrations of a thick periodic junction with concentrated masses on the fine rods, Nonlinear Oscillations, Volume 2 (1999), pp. 511-523

[4] A. Gaudiello Asymptotic behavior of non-homogeneous Neumann problems in domains with oscillating boundary, Ricerche Mat., Volume 43 (1994), pp. 239-292

[5] T.A. Mel'nik; S.A. Nazarov Asymptotic behavior of the solution of the Neumann spectral problem in a domain of “Tooth Comb” Type, J. Math. Sci. (New York), Volume 85 (1997) no. 6, pp. 2326-2346

[6] I. Babuška; R. Vyborny Continuous dependence of eigenvalues on the domains, Czech. Math. J., Volume 15 (1965), pp. 169-178

[7] W. Jäger; A. Mikelić On the roughness-induced effective boundary conditions for an incompressible viscous flow, J. Differential Equations, Volume 170 (2001), pp. 96-122

[8] M. Lobo-Hidalgo; E. Sánchez-Palencia Sur certaines propriétés spectrales des perturbations du domaine dans les problèmes aux limites, Comm. Partial Differential Equations, Volume 4 (1979), pp. 1085-1098

[9] Y. Amirat; G.A. Chechkin; R.R. Gadyl'shin Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with oscillating boundary, Zh. Vychisl. Mat. i Mat. Fiz., Volume 46 (2006) no. 1, pp. 102-115 (Translation in Comput. Math. Math. Phys., 46, 1, 2006, pp. 97-110)

[10] A.M. Il'in Matching of Asymptotic Expansions of Solutions of Boundary-Value Problems, Amer. Math. Soc., Providence, RI, 1992

Cited by Sources:

Comments - Policy