Asymptotics of Nonlocal Energies with a Bilateral Constraint

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