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, >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 -periodic array of small squares of side r_ ^d/d-p. Our analysis is performed in the framework of -convergence and we prove that, in the regime =o (r_), the discrete energy and their continuum counterpart share the same -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.