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

## TL;DR

This paper studies how solutions to a Laplace equation in a domain with two small, closely spaced holes behave as the holes shrink and approach each other, providing an asymptotic expansion of the solution.

## Contribution

It introduces a framework for analyzing the asymptotic behavior of solutions in perforated domains with close holes, using real analytic maps and asymptotic expansions.

## Key findings

- Derived an asymptotic expansion for the solution as holes shrink
- Described the solution behavior in terms of real analytic maps
- Analyzed the effect of hole proximity on the solution

## 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.