A not-so-strange term coming from somewhere

Abstract

We consider Laplace's equation in a periodically perforated domain with Robin boundary conditions on the holes, where the Robin coefficient is scaled proportionally to the inverse total surface area of the performations. We identify a regime in which surface and bulk effects contribute at the same order and show that the homogenised equation contains an additional zeroth-order term depending nonlinearly on the Robin parameter. This term is characterised via a Steklov-type spectral problem in which the spectral parameter appears both in the equation and in the boundary condition. The resulting term interpolates continuously between the Neumann and Dirichlet limits, recovering the classical capacitary strange term in the strong-coupling limit.