We characterize the relaxation of the perimeter in an infinite dimensional Wiener space, with respect to the weak L-topology. We also show that the rescaled AllenCahn functionals approximate this relaxed functional in the sense of Γ-convergence.

The goal of this paper is to solve a long standing open problem, namely, the asymptotic development of order 2 by Γ-convergence of the mass-constrained Cahn–Hilliard functional. This is achieved by introducing a novel rearrangement technique, which works without Dirichlet boundary conditions.

This paper deals with the asymptotic analysis of the three-dimensional problem for a linearly elastic cantilever having an open crosssection which is the union of rectangles with sides of order ε and ε2, as ε goes to zero. Under suitable assumptions on the given loads and for homogeneous and isotropic material, we show that the threedimensional problem Γ-converges to the classical one-dimension...

In this paper we consider the asymptotic behavior of the GinzburgLandau model for superconductivity in 3-d, in various energy regimes. We rigorously derive, through an analysis via Γ-convergence, a reduced model for the vortex density, and deduce a curvature equation for the vortex lines. In the companion paper [2] we describe further applications to superconductivity and superfluidity, such as...

A class of free-discontinuity problems is approximated in the sense of Γ-convergence by a sequence of non-local integral functionals.

In this paper, we study nonlocal gradients and their relationship to classical gradients. As the nonlocality vanishes we demonstrate the convergence of nonlocal gradients to their local analogue for Sobolev and BV functions. As a consequence of these localizations we give new characterizations of the Sobolev and BV spaces that are in the same spirit of Bourgain, Brezis, and Mironsecu’s 2001 cha...

It is well-known that Γ-convergence of functionals provides a tool for studying global and local minimizers. Here we present a general result establishing the existence of critical points of a Γ-converging sequence of functionals provided the associated Γ-limit possesses a nondegenerate critical point, subject to certain mild additional hypotheses. We then go on to prove a theorem that describe...

Abstract. We study the H-norm of the function 1 on tubular neighbourhoods of curves in R. We take the limit of small thickness ε, and we prove two different asymptotic results. The first is an asymptotic development for a fixed curve in the limit ε → 0, containing contributions from the length of the curve (at order ε), the ends (ε), and the curvature (ε). The second result is a Γ-convergence r...

We discuss the Γ-convergence, under the appropriate scaling, of the energy functional ‖u‖Hs(Ω) + ∫ Ω W (u) dx, with s ∈ (0, 1), where ‖u‖Hs(Ω) denotes the total contribution from Ω in the Hs norm of u, and W is a double-well potential. When s ∈ [1/2, 1), we show that the energy Γ-converges to the classical minimal surface functional – while, when s ∈ (0, 1/2), it is easy to see that the functio...

We study, through a Γ-convergence procedure, the discrete to continuum limit of Ising type energies of the form Fε(u) = − ∑ i,j ci,juiuj , where u is a spin variable defined on a portion of a cubic lattice εZ ∩ Ω, Ω being a regular bounded open set, and valued in {−1, 1}. If the constants ci,j are non negative and satisfies suitable coercivity and decay assumptions, we show that all possible Γ-...