Interaction of scales for a singularly perturbed degenerating nonlinear Robin problem

Abstract

We study the asymptotic behaviour of solutions of a boundary value problem for the Laplace equation in a perforated domain in R n , n ≥ 3 , with a (nonlinear) Robin boundary condition on the boundary of the small hole. The problem we wish to consider degenerates in three respects: 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 ϵ of the small hole where we consider the Robin condition collapses to 0. We study how these three singularities interact and affect the asymptotic behaviour as ϵ 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. This article is part of the theme issue ‘Non-smooth variational problems and applications’.