Global representation and multiscale expansion for the Dirichlet problem in a domain with a small hole close to the boundary

Abstract

For each pair (Formula presented.) of positive parameters, we define a perforated domain (Formula presented.) by making a small hole of size (Formula presented.) in an open regular subset (Formula presented.) of (Formula presented.) ((Formula presented.)). The hole is situated at distance (Formula presented.) from the outer boundary (Formula presented.) of the domain. Thus, when (Formula presented.) both the size of the hole and its distance from (Formula presented.) tend to zero, but the size shrinks faster than the distance. Next, we consider a Dirichlet problem for the Laplace equation in the perforated domain (Formula presented.) and we denote its solution by (Formula presented.) Our aim is to represent the map that takes (Formula presented.) to (Formula presented.) in terms of real analytic functions of (Formula presented.) defined in a neighborhood of (0, 0). In contrast with previous results valid only for restrictions of (Formula presented.) to suitable subsets of (Formula presented.) we prove a global representation formula that holds on the whole of (Formula presented.) Such a formula allows us to rigorously justify multiscale expansions, which we subsequently construct.