Comptes Rendus
no. 3
On scattering frequencies in homogenization problems. Critical cases
[Sur les fréquences de diffusion dans les problèmes d'homogénéisation. Cas critiques]
Rustem R. Gadyl'shin1, 2
1 Bashkir State Pedagogical University, 3a October Revolution str., 450000 Ufa, Russia
2 Bashkir State University, 32 Zaki Validi str., 450076 Ufa, Russia
Comptes Rendus. Mécanique, Volume 344 (2016) no. 3, pp. 181-189.

On considère des problèmes aux limites bidimensionnels pour l'équation de Helmholtz avec conditions aux limites de Dirichlet ou de Neumann sur un ensemble d'arcs. Cet ensemble est obtenu à partir d'une courbe fermée en découpant des petits trous situés près les uns des autres, avec une structure localement périodique. Nous construisons le développement asymptotique des fréquences de diffusion (pôles du prolongement analytique des solutions) avec des parties imaginaires petites, qui convergent vers les racines carrées des valeurs propres multiples du problème limite.

We consider a two-dimensional boundary value problems for the Helmholtz equation with Dirichlet and Neumann boundary conditions on a set of arcs. This set is obtained from a closed curve by cutting out small holes situated close to each other and having a locally periodic structure. We construct the asymptotics of the scattering frequencies (poles of the analytic continuation of solutions) with small imaginary parts, which converge to the square roots of multiple eigenvalues of limit problems.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2015.10.002
Keywords: Asymptotics, Small parameter, Homogenization
Mot clés : Asymptotique, Petit paramètre, Homogénéisation
Rustem R. Gadyl'shin 1, 2

1 Bashkir State Pedagogical University, 3a October Revolution str., 450000 Ufa, Russia
2 Bashkir State University, 32 Zaki Validi str., 450076 Ufa, Russia 
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     title = {On scattering frequencies in homogenization problems. {Critical} cases},
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Rustem R. Gadyl'shin. On scattering frequencies in homogenization problems. Critical cases. Comptes Rendus. Mécanique, Volume 344 (2016) no. 3, pp. 181-189. doi : 10.1016/j.crme.2015.10.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2015.10.002/

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