The Dirichlet problem in a planar domain with two moderately close holes

2017/05/05 by Riva, M. Dalla, Musolino, P.
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1705.02142

Abstract

We investigate a Dirichlet problem for the Laplace equation in a domain of ℝ² with two small close holes. The domain is obtained by making in a bounded open set two perforations at distance |ε₁| one from the other and each one of size |ε₁ε₂|. In such a domain, we introduce a Dirichlet problem and we denote by u_ε1,ε2 its solution. We show that the dependence of u_ε1,ε2 upon (ε₁,ε₂) can be described in terms of real analytic maps of the pair (ε₁,ε₂) defined in an open neighborhood of (0,0) and of logarithmic functions of ε₁ and ε₂. Then we study the asymptotic behaviour of of u_ε1,ε2 as ε₁ and ε₂ tend to zero. We show that the first two terms of an asymptotic approximation can be computed only if we introduce a suitable relation between ε₁ and ε₂.

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