A small hole close to the boundary for the two-dimensional Laplace Dirichlet problem
Virginie Bonnaillie‐Noël, Маттео Далла Ріва, Marc Dambrine, Паоло Мусоліно · arXiv (Cornell University) · 2016
We study a Dirichlet problem in a planar domain with a small hole close to the boundary. For each pair e = (e 1 , e 2 ) of positive parameters, we define a perforated domain Ω e obtained by making a small perforation of size e 1 e 2 in an open set. The distance of the cavity from the boundary is instead controlled by e 1 . As e 1 → 0, the perforation shrinks to a point and at the same time approaches the boundary. We consider separately two cases: the case when e tends to (0, 0) and the case when e 1 tends to 0 and e 2 is fixed. In the first case we show that the solution of a Dirichlet problem defined in Ω e displays a logarithmic behavior when e → 0. In the second case instead, the asymptotic behavior of the solution can be described in terms of real analytic functions of e 1 . We will also show that the energy integral and the total flux on the exterior boundary have a different limiting behavior in the two cases.