Comptes Rendus
no. 9
On resonances in a homogenization problem
[Sur les résonances dans un problème d'homogénéisation]

Présenté par : Évariste Sanchez-Palencia

Rustem R. Gadyl'shin1
1 Bashkir State Pedagogical University, 3a October revolution str., 450000 Ufa, Russia
Comptes Rendus. Mécanique, Volume 331 (2003) no. 9, pp. 595-600.

On considère un problème aux limites bidimensionnel pour l'équation de Helmholtz avec des conditions de Neumann sur un ensemble d'arcs. Cet ensemble s'obtient à partir d'une courbe fermée en supprimant des petites parties très proches les unes des autres et disposées de façon périodique. Nous déduisons le comportement asymptotique des fréquences de diffusion (pôles des prolongements analytiques des solutions) avec une petite partie imaginaire et nous montrons qu'elles impliquent l'existence des résonances.

We consider a two-dimensional boundary value problem for the Helmholtz equation with Neumann boundary condition on a set of arcs. This set is obtained from a closed curve by cutting out small holes situated closely each to other and having locally periodic structure. We construct asymptotics of scattering frequencies (poles of analytic continuation of solutions) with small imaginary parts and show that these scattering frequencies imply resonances.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-0721(03)00139-6
Keywords: Vibrations, Asymptotics, Small parameter, Homogenization
Mot clés : Vibrations, Asymptotique, Un petit paramètre, Homogénéisation
Rustem R. Gadyl'shin 1

1 Bashkir State Pedagogical University, 3a October revolution str., 450000 Ufa, Russia 
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Rustem R. Gadyl'shin. On resonances in a homogenization problem. Comptes Rendus. Mécanique, Volume 331 (2003) no. 9, pp. 595-600. doi : 10.1016/S1631-0721(03)00139-6. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(03)00139-6/

[1] V.A. Marchenko; E.Ya. Khruslov Boundary Value Problems in Domains with Fine-Grained Boundaries, Naukova Dumka, Kyiv, 1974 (in Russian)

[2] É. Sanchez-Palencia Boundary value problems in domains containing perforated walls (H. Brezis; J.-L. Lions, eds.), Nonlinear Partial Differential Equation and Their Applications College de France Seminar III, Res. Notes Math., 70, Pitman, London, 1982, pp. 309-325

[3] R.R. Gadyl'shin On analogs of Helmholtz resonator in averaging theory, Mat. Sb., Volume 193 (2002) no. 11, pp. 43-70 (English translation: Sb. Math., 193, 11, 2002, pp. 1611-1638)

[4] A.M. Il'in Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, American Mathematical Society, Providence, RI, 1992

[5] R.R. Gadyl'shin Asymptotics of the eigenvalues of a boundary value problem with rapidly oscillating boundary conditions, Differentsial'nye Uravneniya, Volume 35 (1999) no. 4, pp. 540-551 (English translation: Differential Equations, 35, 4, 1999, pp. 540-551)

[6] R.R. Gadyl'shin Homogenization and asymptotics for a membrane with closely spaced clamping points, Zh. Vychisl. Mat. i Mat. Fiz., Volume 41 (2001) no. 12, pp. 1857-1869 (English translation: Comput. Math. Math. Phys., 41, 12, 2001, pp. 1765-1776)

[7] Lord Rayleigh The theory of Helmholtz resonator, Proc. Roy. Soc. London Ser. A, Volume 92 (1916), pp. 265-275

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