Variational analysis of nonlocal Dirichlet problems in periodically perforated domain

In this paper we consider a family of non local functionals of convolution-type depending on a small parameter and -converging to local functionals defined on Sobolev spaces as . 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 centered on a –periodic lattice, being an additional small parameter and . We highlight differences and analogies with the local case, according to the interplay between the three scales , and . 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.