Interaction of scales for a singularly perturbed degenerating nonlinear Robin problem

Abstract

We study the asymptotic behavior of the 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 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 ǫ 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 ǫ 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.