Asymptotic analysis a perturbed Robin problem in a planar domain

TL;DR

This paper investigates the asymptotic behavior of solutions to a perturbed Robin problem in a planar domain with a small hole, focusing on how boundary conditions influence solutions as the hole size approaches zero.

Contribution

It provides a detailed asymptotic analysis of solutions in a perforated domain with a small hole, considering degenerating Robin boundary conditions.

Findings

As the hole size tends to zero, the solution behavior is significantly affected by the boundary condition degeneracy.

The Robin condition may effectively become a Neumann condition in the limit, depending on the divergence of the Robin datum.

The analysis clarifies the impact of geometric and boundary condition degeneracies on solution asymptotics.

Abstract

We consider a perforated domain $\Omega(\epsilon)$ of $\mathbb{R}^2$ with a small hole of size $\epsilon$ and we study the behavior of the solution of a mixed Neumann-Robin problem in $\Omega(\epsilon)$ as the size $\epsilon$ of the small hole tends to $0$. In addition to the geometric degeneracy of the problem, the $\epsilon$-dependent Robin condition may degenerate into a Neumann condition for $\epsilon=0$ and the Robin datum may diverge to infinity. Our goal is to analyze the asymptotic behavior of the solutions to the problem as $\epsilon$ tends to $0$ and understand how the boundary condition affects the behavior of the solutions when $\epsilon$ is close to $0$.