Asymptotic analysis of perturbed Robin problems in a planar domain

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 \(n\geq 3\). For more information see https://ejde.math.txstate.edu/Volumes/2023/57/abstr.html