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 $e = (e1 ,e2 )$ of positive parameters, we consider a perforated domain $domeps$ obtained by making a small hole of size $e1 e2 $ in an open regular subset $dom$ of $mathbb{R}^n$ at distance $e1$ from the boundary $partialOmega$. As $e1 o 0$, the perforation shrinks to a point and, at the same time, approaches the boundary. When $e o (0,0)$, the size of the hole shrinks at a faster rate than its approach to the boundary. We denote by $\ueps$ the solution of a Dirichlet problem for the Laplace equation in $domeps$. For a space dimension $ngeq 3$, we show that the function mapping $e$ to $\ueps$ has a real analytic continuation in a neighborhood of $(0,0)$. By contrast, for $n=2$ we consider two different regimes: $e$ tends to $(0,0)$, and $e1$ tends to $0$ with $e2$ fixed. When $e o(0,0)$, the solution $\ueps$ has a logarithmic behavior; when only $e1 o0$ and $e2$ is fixed, the asymptotic behavior of the solution can be described in terms of real analytic functions of $e1$. 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.

We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair is an element of = (is an element of(1), is an element of(2)) of positive parameters, we consider a perforated domain & nbsp;Omega(is an element of). obtained by making a small hole of size is an element of(1)is an element of(2) in an open regular subset & nbsp;Omega of R-n at distance is an element of(1) from the boundary partial derivative Omega. As is an element of -> (0, 0), the perforation shrinks to a point and, at the same time, approaches the boundary. When is an element of & nbsp;-> (0, 0), the size of the hole shrinks at a faster rate than its approach to the boundary. We denote by U-is an element of the solution of a Dirichlet problem for the Laplace equation in Omega(is an element of) . For a space dimension n >= 3, we show that the function mapping is an element of to & nbsp; u(is an element of) a real analytic continuation in a neighborhood of (0, 0). By contrast, for n = 2 we consider two different regimes: is an element of tends to (0, 0), and & nbsp;is an element of(1) tends to 0 with is an element of(2) fixed. When is an element of & nbsp;-> (0, 0), the solution u(is an element of) has a logarithmic behavior; when only is an element of(1) -> 0 and is an element of(2) is fixed, the asymptotic behavior of the solution can be described in terms of real analytic functions of is an element of(1). 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. (C) 2018 Elsevier Masson SAS. All rights reserved.