Comptes Rendus
no. 11
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

Virginie Bonnaillie-Noël1 ; Marc Dambrine2 ; Sébastien Tordeux3 ; Grégory Vial1
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, Volume 345 (2007) no. 11, pp. 609-614.

La présence de petites inclusions dans un domaine de référence Ω0 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 ∂Ω0 qui est lui même perturbé.

The presence of small inclusions modifies the solution of the Laplace equation posed in a reference domain Ω0. 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 ∂Ω0.

Reçu le :
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DOI : 10.1016/j.crma.2007.10.037

Virginie Bonnaillie-Noël 1 ; Marc Dambrine 2 ; Sébastien Tordeux 3 ; Grégory Vial 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 
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Virginie Bonnaillie-Noël; Marc Dambrine; Sébastien Tordeux; Grégory Vial. On moderately close inclusions for the Laplace equation. Comptes Rendus. Mathématique, Volume 345 (2007) no. 11, pp. 609-614. doi : 10.1016/j.crma.2007.10.037. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.037/

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