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.

The generalization to gradient vector elds of the classical double-well, singularly perturbed func-tionals, I" (u;) := Z 1 " W(ru) + "jr 2 uj 2 dx; where W() = 0 if and only if = A or = B, and A ? B is a rank-one matrix, is considered. Under suitable constitutive and growth hypotheses on W it is shown that I" ?-converge to +1 otherwise, where K is the (constant) interfacial energy per unit area.

We give a Γ-convergence result for vector-valued nonlinear energies defined on periodically perforated domains. We consider integrands with p-growth for p converging to the space dimension n. We prove that for p close to the critical exponent n there are three regimes, two with a non-trivial size of the perforations (exponential and mixed polynomial-exponential) and one where the Γ-limit is alw...

We study Γ-convergence for a sequence of parabolic function-als, F ε (u) = T 0 Ω f (x ε , t, ∇u)dxdt as ε → 0, where the integrand f is nonconvex, and periodic on the first variable. We obtain the representation formula of the Γ-limit. Our results in this paper support a conclusion which relates Γ-convergence of parabolic functionals to the associated gradient flows and confirms one of De Giorg...

This paper addresses a three-dimensional model for isothermal stress-induced transformation in shape memory polycrystalline materials in presence of permanent inelastic effects. The basic features of the model are recalled and the constitutive and the three-dimensional quasi-static evolution problem are proved to be well-posed. Finally, we discuss the convergence of the model to reduced/former ...