Submitted Paper

Inserted: 23 jun 2024
Last Updated: 23 jun 2024

Year: 2024

Abstract:

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

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