On a singularly perturbed Steklov problem in a domain perforated along the boundary

Gregory A. Chechkin; Ciro D' Apice; Umberto De Maio; Rustem R. Gadyl'shin

doi:10.1016/j.crme.2015.09.001

On a singularly perturbed Steklov problem in a domain perforated along the boundary
[Le problème Steklov singulièrement perturbé dans un domaine perforé le long de la frontière]

Gregory A. Chechkin ¹ ; Ciro D' Apice ² ; Umberto De Maio ³ ; Rustem R. Gadyl'shin ^4,⁵

¹ Department of Differential Equations, Faculty of Mechanics and Mathematics, M.V. Lomonosov Moscow State University, Moscow 119991, Russia
² Dipartimento di Ingegneria dell'Informazione e Matematica Applicata, Università degli Studi di Salerno, Via Ponte don Melillo, 1, Fisciano (SA) 84084, Italia
³ Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli Federico II, Complesso Monte S. Angelo – Edificio “T”, Via Cintia, 80126 Napoli, Italia
⁴ Department of Mathematics and Statistics, Faculty of Physics and Mathematics, Bashkir State Pedagogical University, Ufa 450000, Russia
⁵ Department of Mathematical Analysis, Faculty of Mathematics and IT, Bashkir State University, Ufa 450076, Russia

Comptes Rendus. Mécanique, Volume 344 (2016) no. 1, pp. 12-18.

Résumé
Abstract

Nous étudions le comportement asymptotique des solutions et des éléments propres à un problème aux limites pour l'équation de Laplace dans un domaine perforé le long d'une partie de la frontière. Sur la frontière de trous, nous posons la condition de Dirichlet homogène et la condition spectrale de Steklov sur la part mentionnée de la frontière extérieure du domaine. En supposant que la microstructure de la frontière est périodique, nous construisons le problème aux limites et prouvons le théorème d'homogénéisation.

We study the asymptotic behavior of solutions and eigenelements to a boundary value problem for the Laplace equation in a domain perforated along part of the boundary. On the boundary of holes, we set the homogeneous Dirichlet boundary condition and the Steklov spectral condition on the mentioned part of the outer boundary of the domain. Assuming that the boundary microstructure is periodic, we construct the limit problem and prove the homogenization theorem.

Reçu le : 2015-08-10
Accepté le : 2015-09-11
Publié le : 2015-11-11

DOI : 10.1016/j.crme.2015.09.001

Keywords: Homogenization, Steklov spectral problem, Asymptotic methods
Mot clés : Homogénéisation, Problème spectral de Steklov, Méthodes asymptotiques

Affiliations des auteurs :

Gregory A. Chechkin ¹ ; Ciro D' Apice ² ; Umberto De Maio ³ ; Rustem R. Gadyl'shin ^{4, 5}

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     title = {On a singularly perturbed {Steklov} problem in a domain perforated along the boundary},
     journal = {Comptes Rendus. M\'ecanique},
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Gregory A. Chechkin; Ciro D' Apice; Umberto De Maio; Rustem R. Gadyl'shin. On a singularly perturbed Steklov problem in a domain perforated along the boundary. Comptes Rendus. Mécanique, Volume 344 (2016) no. 1, pp. 12-18. doi : 10.1016/j.crme.2015.09.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2015.09.001/

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