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

Matteo Dalla Riva, Paolo Musolino

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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 uϵ1,ϵ2 its solution. We show that the dependence of uϵ1,ϵ2 upon (ϵ1,ϵ2) can be described in terms of real analytic maps of the pair (ϵ1,ϵ2) defined in an open neighbourhood of (0,0) and of logarithmic functions of ϵ1 and ϵ2. Then we study the asymptotic behaviour of uϵ1,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.
Original languageEnglish
Pages (from-to)2567-2605
Number of pages39
JournalJournal of Differential Equations
Issue number5
Early online date12 Apr 2017
Publication statusPublished - 05 Sept 2017


  • Dirichlet problem
  • singularly perturbed perforated planar domain
  • moderately close holes
  • laplace operator
  • real analytic continuation in Banach space
  • asymptotic expansion


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