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

We study the asymptotic behavior, as the mesh size tends to zero, of a general class of discrete energies defined on functions u: ZN\ u () Rᵈ of the form eqnarray* F_ (u) = ₀ₑₑ₀ₘ₋₋ {, ZN \\ 91, 93 array }-0. 3cm g_ (, , u () -u () ) eqnarray* and satisfying superlinear growth conditions. We show that all the possible varia\-tional limits are defined on W^1, p (;{ Rᵈ}) of the local type _ 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).