---
author:
- 'Matteo Dalla Riva [^1]  and Paolo Musolino[^2]'
date:  
title: 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.

[**Keywords:**]{} mixed problem; singularly perturbed perforated domain; moderately close holes; Laplace operator; real analytic continuation in Banach space; asymptotic expansion

[[**2010 Mathematics Subject Classification:**]{}]{} 35J25; 31B10; 45A05; 35B25; 35C20

Introduction {#introd}
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The analysis of singular domain perturbation problems for linear equations and system of partial differential equations has caught the attention of several authors. In particular, a wide literature has been dedicated to the study of boundary value problems defined in domains 