Singular behavior for a multi-parameter periodic Dirichlet problem

TL;DR

This paper investigates how solutions to a periodic Dirichlet problem for the Poisson equation behave as the holes in the domain become very small, revealing singular behaviors especially when the holes shrink to points.

Contribution

It introduces a new analysis of the solution's dependence on small hole size without assuming zero integral of the Poisson data, highlighting singular behaviors in different dimensions.

Findings

Solution can be expressed as an analytic map times a singular factor involving epsilon

For dimensions ≥ 3, the solution scales as 1/epsilon^{n-2}

In dimension 2, a logarithmic term appears in the solution representation

Abstract

We consider a Dirichlet problem for the Poisson equation in a periodically perforated domain. The geometry of the domain is controlled by two parameters: a real number $\epsilon>0$ proportional to the radius of the holes and a map $\phi$, which models the shape of the holes. So, if $g$ denotes the Dirichlet boundary datum and $f$ the Poisson datum, we have a solution for each quadruple $(\epsilon,\phi,g,f)$. Our aim is to study how the solution depends on $(\epsilon,\phi,g,f)$, especially when $\epsilon$ is very small and the holes narrow to points. In contrast with previous works, we don't introduce the assumption that $f$ has zero integral on the fundamental periodicity cell. This brings in a certain singular behavior for $\epsilon$ close to $0$. We show that, when the dimension $n$ of the ambient space is greater than or equal to $3$, a suitable restriction of the solution can be represented with an analytic map of the quadruple $(\epsilon,\phi,g,f)$ multiplied by the factor $1/\epsilon^{n-2}$. In case of dimension $n=2$, we have to add $\log \epsilon$ times the integral of $f/2\pi$.