A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary

We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair ε=(ε1,ε2) of positive parameters, we consider a perforated domain Ωε obtained by making a small hole of size ε1ε2 in an open regular subset Ω of Rn at distance ε1 from the boundary ∂Ω. As ε1→0, the perforation shrinks to a point and, at the same time, approaches the boundary. When ε→(0,0), the size of the hole shrinks at a faster rate than its approach to the boundary. We denote by uε the solution of a Dirichlet problem for the Laplace equation in Ωε. For a space dimension n≥3, we show that the function mapping ε to uε has a real analytic continuation in a neighborhood of (0,0). By contrast, for n=2 we consider two different regimes: ε tends to (0,0), and ε1 tends to 0 with ε2 fixed. When ε→(0,0), the solution uε has a logarithmic behavior; when only ε1→0 and ε2 is fixed, the asymptotic behavior of the solution can be described in terms of real analytic functions of ε1. We also s...