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arXiv · 2211.11631

Singular behavior for a multi-parameter periodic Dirichlet problem

Matteo Dalla Riva·Paolo Luzzini·Paolo Musolino
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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 $ε>0$ proportional to the radius of the holes and a map $ϕ$, 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 $(ε,ϕ,g,f)$. Our aim is to study how the solution depends on $(ε,ϕ,g,f)$, especially when $ε$ 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 $ε$ 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 $(ε,ϕ,g,f)$ multiplied by the factor $1/ε^{n-2}$. In case of dimension $n=2$, we have to add $\log ε$ times the integral of $f/2π$.

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Matteo Dalla Riva, Paolo Luzzini, Paolo Musolino. 2022-11-21. Singular behavior for a multi-parameter periodic Dirichlet problem. https://arxiv.org/abs/2211.11631

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