Interactions between moderately close circular inclusions: The Dirichlet–Laplace equation in the plane

Abstract

The presence of small inclusions or of a surface defect modifies the solution of the Laplace equation posed in a reference domain Ω₀. If the characteristic size of the perturbation is small, then one can expect that the solution of the problem posed on the perturbed geometry is close to the solution of the reference shape. Asymptotic expansion with respect to that small parameter – the characteristic size of the perturbation – can then be performed. We consider in the present work the case of two circular defects with homogeneous Dirichlet boundary conditions in a bidimensional domain, we distinguish the cases where the distance between the object is of order 1 and the case where it is larger than the characteristic size of the defects but small with respect to the size of the domain. In both cases, we derive the complete expansion and provide some numerical illustrations.