this discrete model can be approximated by a continuous energy defined on special functions with bounded variation. In fact, if we limit the interactions in the sum to the nearest neighbors in the portion of εZ contained in some fixed Ω, and we interpret the values ui as the discretization of a function defined in Ω, then these energies can be studied using the methods of Γ-convergence, and the...

We study the asymptotic behavior of functionals associated to the energy of a thin nonlinear elastic spherical shell in the limit of vanishing thickness (proportional to a small parameter) ε and under the assumption of radial deformations. The functionals are characterized by the presence of a nonlocal potential term and defined on suitable weighted functional spaces. The transition shell-membr...

In this paper, we study the relation between the second inner variations of the Allen-Cahn functional and its Gamma-limit, the area functional. Our result implies that the Allen-Cahn functional only approximates well the area functional up to the first order. However, as an application of our result, we prove, assuming the singlemultiplicity property of the limiting energy, that the Morse indic...

We present the second of two articles on the small volume-fraction limit of a nonlocal Cahn–Hilliard functional introduced to model microphase separation of diblock copolymers. After having established the results for the sharp-interface version of the functional [SIAM J. Math. Anal., 42 (2010), pp. 1334–1370], we consider here the full diffuse-interface functional and address the limit in whic...

The purpose of this paper is to investigate the Cahn-Hillard approximation for entire minimal hypersurfaces in the hyperbolic space. Combining comparison principles with minimization and blow-up arguments, we prove existence results for entire local minimizers with prescribed behaviour at infinity. Then, we study the limit as the length scale tends to zero through a Γ-convergence analysis, obta...

We study compactness and Γ-convergence for Ginzburg-Landau type functionals. We only assume that the potential is continuous and positive definite close to one circular well, but allow large zero sets inside the well. We show that the relaxation of the assumptions does not change the results to leading order unless the energy is very large.

A homogenization result is given for a material having soft inclusions arranged in a periodic structure, under the requirement that the interpenetration of matter is forbidden. According to the relation between the softness parameter and the size of the microstucture, three different limit models are deduced via Γ-convergence.

This paper is an extended version of the lecture delivered at the Summer School on Differential Equations and Calculus of Variations (Pisa, September 16-28, 1996). That lecture was conceived as an introduction to the theory of Γ-convergence and in particular to the Modica-Mortola theorem; I have tried to reply the style and the structure of the lecture also in the written version. Thus first co...

The eeective energy of a mixture of two elastic materials in a thin lm is characterized using Gamma-limit techniques. For cylindrical shaped inclusions it is shown that 3D-2D asymptotics and optimal design commute from a variational viewpoint. Regularity of local minimizers for the resulting design is addressed.