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

We investigate a Dirichlet problem for the Laplace equation in a domain of $\mathbb{R}^2$ with two small close holes. The domain is obtained by making in a bounded open set two perforations at distance $|\epsilon_1|$ one from the other and each one of size $|\epsilon_1\epsilon_2|$. In such a domain, we introduce a Dirichlet problem and we denote by $u_{\epsilon_1,\epsilon_2}$ its solution. We show that the dependence of $u_{\epsilon_1,\epsilon_2}$ upon $(\epsilon_1,\epsilon_2)$ can be described in terms of real analytic maps of the pair $(\epsilon_1,\epsilon_2)$ defined in an open neighborhood of $(0,0)$ and of logarithmic functions of $\epsilon_1$ and $\epsilon_2$. Then we study the asymptotic behaviour of of $u_{\epsilon_1,\epsilon_2}$ as $\epsilon_1$ and $\epsilon_2$ 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 $\epsilon_1$ and $\epsilon_2$.