A general integral representation result for continuum limits of discrete energies with superlinear growth

For any given bounded open set $\Omega$, we study the asymptotic behavior, as the mesh size $\e$ tends to zero, of a general class of discrete pairwise interaction energies $F_\e$. Under natural growth and coercivity hypotheses on the dependence of such energies on difference quotients we show that all the possible variational limits of $F_\e$ are defined on $W^{1,p}(\Omega;\rd)$ and are of the type $$ \int_\Om f(x,\nabla u)\, dx. $$ We also show that in general $f$ may be a quasiconvex non convex function even if very simple interactions are considered. We also address the case of homogenization giving a general asymptotic formula that can be simplified in many situations (e.g. in the case of nearest neighbor interactions or under convexity hypotheses).