A General Integral Representation Result for Continuum Limits of Discrete Energies with Superlinear Growth

We study the asymptotic behavior, as the mesh size ε tends to zero, of a general class of discrete energies defined on functions u:α∈ε ZN∩\ Ω↦ u(α)∈ Rᵈ of the form {eqnarray*} Fε(u)=∑_{ {array}{ll} { α, β ∈ ε Z^N} \\ {91}α ,β {93} ⊂Ω {array} }-0.3cm gε(α,β,u(α)-u(β)) {eqnarray*} and satisfying superlinear growth conditions. We show that all the possible varia\-tional limits are defined on W1,p(Ω; Rᵈ) of the local type ∫_Ω f(x,∇ u)\, dx. We show that, in general, f may be a quasi-convex nonconvex function even if very simple interactions are considered. We also treat 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).