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 >0 and -converging to local functionals defined on Sobolev spaces as 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_ centered on a --periodic lattice, being > 0 an additional small parameter and r_=o (). We highlight differences and analogies with the local case, according to the interplay between the three scales, and r_. 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.