Variational analysis of nonlocal Dirichlet problems in periodically perforated domains

In this paper we consider a family of non local functionals of convolution-type depending on a small parameter $\epsilon >0$ and $\mathrm{\Gamma }$-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 $r_{\delta }$ 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 $\epsilon$, $\delta$ 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.