Locally periodic thin domains with varying period

José M. Arrieta; Manuel Villanueva-Pesqueira

doi:10.1016/j.crma.2014.03.014

Partial differential equations

Locally periodic thin domains with varying period
[Domaines minces localement périodiques avec une période variable]

Présenté par : the Editorial Board

José M. Arrieta ^1,² ; Manuel Villanueva-Pesqueira ¹

¹ Dept. Matemática Aplicada, Facultad Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
² Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, C/Nicolás Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain

Comptes Rendus. Mathématique, Volume 352 (2014) no. 5, pp. 397-403.

Résumé
Abstract

Nous analysons le comportement des solutions de l'équation de Laplace dans un domaine mince à frontière fortement oscillante. Les oscillations sont localement périodiques dans le sens où l'amplitude et la période ne sont pas nécessairement constantes, puisqu'elles varient en espace. Nous obtenons le problème limite homogénéisé et montrons les résultats des correcteurs. Pour atteindre notre objectif, nous étendons la méthode de l'opérateur d'éclatement à des cas localement périodiques. Les idées principales de cette extension peuvent être appliquées à d'autres cas, comme par exemple à des domaines perforés ou à des structures réticulées localement périodiques dont la période n'est pas forcément constante.

We analyze the behavior of the solutions of the Laplace equation with Neumann boundary conditions in a thin domain with a highly oscillatory behavior. The oscillations are locally periodic in the sense that both the amplitude and the period of the oscillations may not be constant and actually they vary in space. We obtain the asymptotic homogenized limit and provide some correctors. To accomplish this goal, we extend the unfolding operator method to the locally periodic case. The main ideas of this extension may be applied to other cases like perforated domains or reticulated structures, which are locally periodic with not necessarily a constant period.

Reçu le : 2013-07-09
Accepté le : 2014-03-17
Publié le : 2014-04-02

DOI : 10.1016/j.crma.2014.03.014

Affiliations des auteurs :

José M. Arrieta ^{1, 2} ; Manuel Villanueva-Pesqueira ¹

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     title = {Locally periodic thin domains with varying period},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {397--403},
     publisher = {Elsevier},
     volume = {352},
     number = {5},
     year = {2014},
     doi = {10.1016/j.crma.2014.03.014},
     language = {en},
}

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José M. Arrieta; Manuel Villanueva-Pesqueira. Locally periodic thin domains with varying period. Comptes Rendus. Mathématique, Volume 352 (2014) no. 5, pp. 397-403. doi : 10.1016/j.crma.2014.03.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.03.014/

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