A mixed problem for the Laplace operator in a domain with moderately close holes

Abstract

We investigate the behavior of the solution of a mixed problem in a domain with two moderately close holes. We introduce a positive parameter $\epsilon$ and we define a perforated domain ${\Omega }_{\epsilon }$ obtained by making two small perforations in an open set. Both the size and the distance of the cavities tend to $0$ as $\epsilon \to 0$. For $\epsilon$ small, we denote by $u_{\epsilon }$ the solution of a mixed problem for the Laplace equation in ${\Omega }_{\epsilon }$. We describe what happens to $u_{\epsilon }$ as $\epsilon \to 0$ in terms of real analytic maps and we compute an asymptotic expansion.

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