On moderately close inclusions for the Laplace equation

Bonnaillie-Noël, Virginie; Dambrine, Marc; Tordeux, Sébastien; Vial, Grégory

doi:10.1016/j.crma.2007.10.037

Partial Differential Equations

On moderately close inclusions for the Laplace equation
[Interactions entre inclusions relativement proches pour l'équation de Laplace]

Présenté par : Philippe G. Ciarlet

Bonnaillie-Noël, Virginie ¹ ; Dambrine, Marc ² ; Tordeux, Sébastien ³ ; Vial, Grégory ¹

¹ IRMAR, ENS Cachan Bretagne, CNRS, UEB, avenue Robert-Schuman, 35170 Bruz, France
² LMAC, Université de technologie de Compiègne, 60200 Compiègne, France
³ MIP, INSA Toulouse, 135, avenue de Rangueil, 31077 Toulouse cedex 4, France

Comptes Rendus. Mathématique, Tome 345 (2007) no. 11, pp. 609-614.

Résumé
Abstract

La présence de petites inclusions dans un domaine de référence $Ω_{0}$ modifie la solution de l'équation de Laplace dans ce domaine. Les cas d'une inclusion isolée ou de plusieurs bien séparées ont été largement étudiés. Dans cette Note, nous considérons le cas où la distance entre deux inclusions tend vers zéro mais reste grande par rapport à leur taille caractéristique. Nous donnons un développement asymptotique multi-échelle complet de la solution de l'équation de Laplace dans la situation de deux inclusions parfaitement isolantes. Nous présentons également le cas d'une seule inclusion proche du bord $\partial Ω_{0}$ qui est lui même perturbé.

The presence of small inclusions modifies the solution of the Laplace equation posed in a reference domain $Ω_{0}$ . This question has been widely studied for a single inclusion or well-separated inclusions. We investigate in this Note the case where the distance between the holes tends to zero but remains large with respect to their characteristic size. We first consider two perfectly insulated inclusions. In this configuration we give a complete multiscale asymptotic expansion of the solution to the Laplace equation. We also address the situation of a single inclusion close to a singular perturbation of the boundary $\partial Ω_{0}$ .

Reçu le : 2007-05-05
Accepté le : 2007-10-22
Publié le : 2007-11-26

DOI : 10.1016/j.crma.2007.10.037

Affiliations des auteurs :

Bonnaillie-Noël, Virginie ¹ ; Dambrine, Marc ² ; Tordeux, Sébastien ³ ; Vial, Grégory ¹

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     author = {Bonnaillie-No\"el, Virginie and Dambrine, Marc and Tordeux, S\'ebastien and Vial, Gr\'egory},
     title = {On moderately close inclusions for the {Laplace} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {609--614},
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     url = {http://www.numdam.org/articles/10.1016/j.crma.2007.10.037/}
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Bonnaillie-Noël, Virginie; Dambrine, Marc; Tordeux, Sébastien; Vial, Grégory. On moderately close inclusions for the Laplace equation. Comptes Rendus. Mathématique, Tome 345 (2007) no. 11, pp. 609-614. doi : 10.1016/j.crma.2007.10.037. http://www.numdam.org/articles/10.1016/j.crma.2007.10.037/

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