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

Abstract

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 show that for n=2, the energy integral and the total flux on the exterior boundary have different limiting values in the two regimes. We prove these results by using functional analysis methods in conjunction with certain special layer potentials.