Variational Analysis of Discrete Dirichlet Problems in Periodically Perforated Domains

In this paper we study the asymptotic behavior of a family of discrete functionals as the lattice size, ε > 0, tends to zero. We consider pairwise interaction energies satisfying p-growth conditions, p < d, d being the dimension of the reference configuration, defined on discrete functions subject to Dirichlet conditions on a δ-periodic array of small squares of side rδ ~ δd/d−p. Our analysis is performed in the framework of Γ-convergence and we prove that, in the regime ε = o(rδ), the discrete energy and their continuum counterpart share the same Γ-limit and the effect of the constraints leads to a capacitary term in the limit energy as in the classical theory of periodically perforated domains for local integral functionals.