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 Ω 0 . 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.

A numerical approach for the Poisson equation in a planar domain with a small inclusion

“…In particular, it can be chosen arbitrarily small. Looking at the explicit definition ofû δ given by (6), this implies the following result.…”

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

“…wω(xε)−ln ε is analogous to those appearing in the case of circular inclusion in [1,Relation (2.6)].…”

“…Nevertheless, estimates in the energy norm in the full domain can be obtained following the strategy used in [2,1] where one decomposes the correctors on homogeneous harmonic functions. Using [2, Proposition 3.2], traces of functions are estimated on the singular boundary ∂ω ε .…”

“…2. The second case presents a third scale η(ε) between ε and 1 such that A typical choice for η(ε) is ε α with α ∈ [0, 1) as made in [2,1].…”

“…The purpose of this paper is to get the lower order terms in the expansion of u ε . We follow the leading ideas of [1] and generalize their results to any geometry of the defects while simplifying the proofs. We also relax the assumption on the regularity of f and do not more assume that f is supported far away the inclusions.…”