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