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 ∈ εZ N ∩ Ω → u(α) ∈ R d of the formand satisfying superlinear growth conditions. We show that all the possible variational limits are defined on W 1,p (Ω; R d ) of the local typeWe 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).

We provide a variational description of nearest-neighbours and next-to-nearest neighbours binary lattice systems. By studying the Γ-limit of proper scaling of the energies of the systems, we highlight phase and antiphase boundary phenomena and show how they depend on the geometry of the lattice.

We introduce a new variational method for the study of stability in the isoperimetric inequality. The method is quite general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter. Two applications are presented. First we give a new proof of the sharp quantitative isoperimetric inequality in R n . Second we positively answer to a conjecture by Hall concerning the best constant for the quantitative isoperimetric inequality in R 2 in the small asymmetry regime.2000 Mathematics Subject Classification. 52A40 (28A75, 49J45).

This article is devoted to the study of the asymptotic behavior of a class of energies defined on stochastic lattices. Under polynomial growth assumptions, we prove that the energy functionals Fε stored in the deformation of an ε-scaling of a stochastic lattice Γ-converge to a continuous energy functional when ε goes to zero. In particular, the limiting energy functional is of integral type, and deterministic if the lattice is ergodic. We also generalize to systems and nonlinear settings well-known results on stochastic homogenization of discrete elliptic equations. As an application of the main result, we prove the convergence of a discrete model for rubber towards the nonlinear theory of continuum mechanics. We finally address some mechanical properties of the limiting models, such as frame-invariance, isotropy and natural states.

We introduce and discuss discrete two-dimensional models for XY spin systems and screw dislocations in crystals. We prove that, as the lattice spacing ε tends to zero, the relevant energies in these models behave like a free energy in the complex Ginzburg-Landau theory of superconductivity, justifying in a rigorous mathematical language the analogies between screw dislocations in crystals and vortices in superconductors. To this purpose, we introduce a notion of asymptotic variational equivalence between families of functionals in the framework of Γ-convergence. We then prove that, in several scaling regimes, the complex Ginzburg-Landau, the XY spin system and the screw dislocation energy functionals are variationally equivalent. Exploiting such an equivalence between dislocations and vortices, we can show new results concerning the asymptotic behavior of screw dislocations in the | log ε| 2 energetic regime.