A strange term coming from the boundary data

TL;DR

This paper investigates the asymptotic behavior of solutions to Poisson's equation in perforated domains with Robin boundary conditions, revealing a homogenized limit that generalizes previous homogeneous Robin results.

Contribution

It extends the analysis of Poisson's equation in perforated domains to inhomogeneous Robin boundary conditions, providing a generalized limit and a homogenized Helmholtz equation.

Findings

Derived a generalized limit for non-periodic domains with arbitrary boundary data.

Showed that for periodic spheres, the homogenized limit satisfies a Helmholtz equation with an extra term.

Extended previous work on homogeneous Robin boundary conditions to inhomogeneous cases.

Abstract

In this paper, I derive the limiting behaviour of the solutions to Poisson's equation, in a perforated domain, subject to inhomogeneous Robin boundary conditions. In the first half of the paper, I derive a generalised limit for non-periodic domains and arbitrary boundary data. In the second half of this paper, I demonstrate that for periodically arranged spheres and identical Robin boundary data on each sphere, the homogenised limit of Poisson's equation satisfies the Helmholtz equation with an additional term in the domain data, which represents the contribution from the inhomogeneous Robin boundary data. These results are a generalisation of the work of Kaizu, who derived the limit of the solutions to the homogeneous Robin problem.