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 \(\varepsilon \rightarrow 0\) . 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.
