Asymptotics of Nonlocal Energies with a Bilateral Constraint

We consider a family of nonlocal functionals

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 ε → 0

, accounting for a possible penalization term of capacitary type.