A not-so-strange term coming from somewhere

TL;DR

This paper analyzes Laplace's equation in a perforated domain with Robin boundary conditions, deriving a homogenized equation that includes a nonlinear term depending on the Robin parameter, bridging Neumann and Dirichlet cases.

Contribution

It identifies a regime where surface and bulk effects are balanced and characterizes a new nonlinear term in the homogenized equation via a Steklov spectral problem.

Findings

The homogenized equation contains a nonlinear term depending on the Robin parameter.

The nonlinear term interpolates between Neumann and Dirichlet limits.

Recovers the classical capacitary strange term in the strong-coupling limit.

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.