Variational analysis of nonlocal Dirichlet problems in periodically perforated domains

Abstract

In this paper we consider a family of non local functionals of convolution-type depending on a small parameter $\epsilon >0$ and $\hat{I}“$-converging to local functionals defined on Sobolev spaces as $\epsilon \to 0$. 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 $r_{\hat{I}}´$ centered on a $\hat{I}´$--periodic lattice, being $\hat{I}´>0$ an additional small parameter and $r_{\hat{I}}´=o(\hat{I}´)$. We highlight differences and analogies with the local case, according to the interplay between the three scales $\epsilon$, $\hat{I}´$ and $r_{\hat{I}}´$. 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.