Asymptotics of Nonlocal Energies with a Bilateral Constraint

We consider a family of nonlocal functionals F_ε F ε depending on a small parameter ε>0 ε > 0 and impose a generic Dirichlet constraint on a periodic array of small balls of radius r_δ r δ centered on a δ δ -periodic lattice. We investigate the asymptotic behaviour of this type of constrained functionals when the order parameter ε ε tends to zero. Such analysis is deeply affected by the interplay between the three scales ε ε , δ_ε δ ε and rδ_ε r δ ε . Assuming that ε=o(rδ_ε) ε = o ( r δ ε ) we perform a complete analysis when the datum assigned on the perforation coincides with a function in C^1( R^d; 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 Γ Γ -convergence of F_ε F ε , with respect to the strong L^p L p topology, to some local functionals defined on Sobolev spaces as ε→ 0 <