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

For each pair epsilon = (epsilon(1), epsilon(2)) of positive parameters, we define a perforated domain Xe by making a small hole of size e1e2 in an open regular subset X of R-n (n >= 3). The hole is situated at distance e1 from the outer boundary partial derivative Omega of the domain. Thus, when epsilon -> (0,0) both the size of the hole and its distance from partial derivative Omega 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 Omega(e) and we denote its solution by u(epsilon): Our aim is to represent the map that takes e to u(epsilon) in terms of real analytic functions of epsilon defined in a neighborhood of (0, 0). In contrast with previous results valid only for restrictions of u(epsilon) to suitable subsets of Omega(e), we prove a global representation formula that holds on the whole of Omega(e): Such a formula allows us to rigorously justify multiscale expansions, which we subsequently construct.