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, $\varepsilon>0$, 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 $\delta$-periodic array of small squares of side $r_{\delta}\sim \delta^{d/d-p}$. Our analysis is performed in the framework of $\Gamma$-convergence and we prove that, in the regime $\varepsilon=o(r_{\delta})$, the discrete energy and their continuum counterpart share the same $\Gamma$-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.