[Approximation asymptotique des éléments propres du problème de Dirichlet pour le Laplacien dans un domaine à frontière fortement oscillante]
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
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.
Accepté le :
Publié le :
Mots-clés : Développement asymptotique, Problème spectral, Frontière oscillante
Youcef Amirat 1 ; Gregory A. Chechkin 2, 3 ; Rustem R. Gadyl'shin 4
@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/
[1] Asymptotics for eigenelements of Laplacian in domain with oscillating boundary: Multiple eigenvalues, Appl. Anal., Volume 86 (2007) no. 7, pp. 873-897
[2] 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] On free vibrations of a thick periodic junction with concentrated masses on the fine rods, Nonlinear Oscillations, Volume 2 (1999), pp. 511-523
[4] Asymptotic behavior of non-homogeneous Neumann problems in domains with oscillating boundary, Ricerche Mat., Volume 43 (1994), pp. 239-292
[5] 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] Continuous dependence of eigenvalues on the domains, Czech. Math. J., Volume 15 (1965), pp. 169-178
[7] On the roughness-induced effective boundary conditions for an incompressible viscous flow, J. Differential Equations, Volume 170 (2001), pp. 96-122
[8] Sur certaines propriétés spectrales des perturbations du domaine dans les problèmes aux limites, Comm. Partial Differential Equations, Volume 4 (1979), pp. 1085-1098
[9] 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] Matching of Asymptotic Expansions of Solutions of Boundary-Value Problems, Amer. Math. Soc., Providence, RI, 1992
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