Asymptotics of Nonlocal Energies with a Bilateral Constraint

We consider a family of nonlocal functionals $F_\varepsilon$ depending on a small parameter $\varepsilon>0$ and impose a generic Dirichlet constraint on a periodic array of small balls of radius $r_\delta$ centered on a $\delta$-periodic lattice. We investigate the asymptotic behaviour of this type of constrained functionals when the order parameter $\varepsilon$ tends to zero. Such analysis is deeply affected by the interplay between the three scales $\varepsilon$, $\delta_\varepsilon$ and $r_{\delta_\varepsilon}$. Assuming that $\varepsilon=o(r_{\delta_\varepsilon})$ we perform a complete analysis when the datum assigned on the perforation coincides with a function in $C^1(\mathbb R^d;\mathbb R^m)$. More precisely, according to the scaling among the periodicity and the radius of the ‘perforation’, we prove the $\Gamma$-convergence of $F_\varepsilon$, with respect to the strong $L^p$ topology, to some local functionals defined on Sobolev spaces as $\varepsilon\to 0$, accounting for a possible penalization term of capacitary type.