Interactions Between Moderately Close Inclusions for the Two-Dimensional Dirichlet–Laplacian

Virginie Bonnaillie-Noël (1) , Marc Dambrine (2) , Christophe Lacave (3)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
3 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

This paper concerns the asymptotic expansion of the solution of the Dirichlet-Laplace problem in a domain with small inclusions. This problem is well understood for the Neumann condition in dimension greater than two or Dirichlet condition in dimension greater than three. The case of two circular inclusions in a bidimensional domain was considered in [1]. In this paper, we generalize the previous result to any shape and relax the assumptions of regularity and support of the data. Our approach uses conformal mapping and suitable lifting of Dirichlet conditions. We also analyze configurations with several scales for the distance between the inclusions (when the number is larger than 2).

Mots clés: perforated domain; 35J05; Dirichlet boundary conditions; asymptotic expansion; conformal mapping; 35C20; 35J08; 35B25

Domaines: Equations aux dérivées partielles [math.AP]

Virginie Bonnaillie-Noël, Marc Dambrine, Christophe Lacave. Interactions Between Moderately Close Inclusions for the Two-Dimensional Dirichlet–Laplacian. Applied Mathematics Research eXpress, 2016, 2016 (1), pp.1 - 23. ⟨10.1093/amrx/abv008⟩. ⟨hal-01120173v2⟩