Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 57, pp. 1-21.

Title: Asymptotic analysis of perturbed Robin problems in a planar domain
       
Authors: Paolo Musolino (Univ. Ca' Foscari Venezia, Venezia Mestre, Italy)
         Martin Dutko (Rockfield Software Limited, Swansea, Wales UK)
         Gennady Mishuris (Aberystwyth Univ., Ceredigion, Aberystwyth, Wales, UK)
         
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 nonlinear $\epsilon$-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 to understand how the boundary condition affects the behavior of the solutions when $\epsilon$ is close to $0$. The present paper extends to the planar case the results of [36]
dealing with the case of dimension $n\geq 3$.

Submitted February 15, 2023. Published September 11, 2023.
Math Subject Classifications: 35J25, 31B10, 35B25, 35C20, 47H30.
Key Words: Singularly perturbed boundary value problem;  Laplace equation; nonlinear Robin condition; perforated planar domain; integral equation