Variational analysis of nonlocal Dirichlet problems in periodically perforated domains

TL;DR

This paper investigates the asymptotic behavior of nonlocal convolution-type functionals in perforated domains as parameters tend to zero, revealing differences from local cases and introducing a novel nonlocal inequality.

Contribution

It introduces a nonlocal Gagliardo-Nirenberg-Sobolev inequality and analyzes the interplay of multiple small scales in nonlocal Dirichlet problems.

Findings

Identifies asymptotic limits of nonlocal functionals in perforated domains.

Highlights differences between nonlocal and local cases.

Develops a new nonlocal inequality potentially useful for other studies.

Abstract

In this paper we consider a family of non local functionals of convolution-type depending on a small parameter $\varepsilon>0$ and $\Gamma$-converging to local functionals defined on Sobolev spaces as $\varepsilon\to 0$. We study the asymptotic behaviour of the functionals when the order parameter is subject to Dirichlet conditions on a periodically perforated domains, given by a periodic array of small balls of radius $r_\delta$ centered on a $\delta$--periodic lattice, being $\delta > 0$ an additional small parameter and $r_\delta=o(\delta)$. We highlight differences and analogies with the local case, according to the interplay between the three scales $\varepsilon$, $\delta$ and $r_\delta$. A fundamental tool in our analysis turns out to be a non local variant of the classical Gagliardo-Nirenberg-Sobolev inequality in Sobolev spaces which may be of independent interest and useful for other applications.