In the absence of constraints, these energies are known to converge, as the interaction scale ε tends to zero, to local functionals of Sobolev type. The aim of this work is to understand how this limit is modified by the presence of microscopic pinning sites and by the interaction between the three relevant scales of the problem: the interaction length ε, the lattice spacing δ, and the size of the perforations (r_δ).

We show that the Γ-limit depends crucially on the relative scaling between these parameters. When the interaction scale is much smaller than the perforation size, the limit coincides with the classical local capacitary correction known for Dirichlet energies in perforated domains. In contrast, when the interaction scale is comparable with the size of the perforations, the limit retains a memory of the nonlocal structure and leads to a nonlocal capacitary term. Finally, when the interaction scale is much larger than the perforations, the pinning sites become asymptotically invisible.

The analysis combines techniques from Γ-convergence, homogenization, and the study of capacitary problems, together with a nonlocal variant of the Gagliardo–Nirenberg–Sobolev inequality that may be of independent interest.