Interaction of scales for a singularly perturbed degenerating nonlinear Robin problem

TL;DR

This paper investigates the complex asymptotic behavior of solutions to a Laplace boundary value problem with nonlinear Robin conditions on a small perforation, analyzing how degenerations in boundary conditions and hole size influence solutions.

Contribution

It provides a detailed analysis of the interaction between multiple singularities in a nonlinear Robin problem and characterizes solutions using real analytic maps as the perforation shrinks.

Findings

Asymptotic representation of solutions in terms of real analytic functions

Characterization of energy integrals under singular perturbations

Insights into the interplay of boundary condition degenerations

Abstract

We study the asymptotic behavior of the solutions of a boundary value problem for the Laplace equation in a perforated domain in $\mathbb{R}^n$, $n\geq 3$, with a (nonlinear) Robin boundary condition on the boundary of the small hole. The problem we wish to consider degenerates under three aspects: in the limit case the Robin boundary condition may degenerate into a Neumann boundary condition, the Robin datum may tend to infinity, and the size $\epsilon$ of the small hole where we consider the Robin condition collapses to $0$. We study how these three singularities interact and affect the asymptotic behavior as $\epsilon$ tends to $0$, and we represent the solution and its energy integral in terms of real analytic maps and known functions of the singular perturbation parameters.