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-...

The distributional k-dimensional Jacobian of a map u in the Sobolev space W 1,k−1 which takes values in the the sphere S can be viewed as the boundary of a rectifiable current of codimension k carried by (part of) the singularity of u which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary M o...

We study the asymptotic behavior of a sequence of Dirichlet problems for second order linear operators in divergence form { −div(σε∇u) = f in Ω, u ∈ H1 0 (Ω), where (σε) ⊂ L∞(Ω;Rn×n) is uniformly elliptic and possibly non-symmetric. On account of the variational principle of Cherkaev and Gibiansky [1], we are able to prove a variational characterization of the H-convergence of (σε) in terms of ...

We review recent results on the homogenization in Sobolev spaces with variable exponents. In particular, we are dealing with the Γ-convergence of variational functionals with rapidly oscillating coefficients, the homogenization of the Dirichlet and Neumann variational problems in strongly perforated domains, as well as double porosity type problems.The growth functions also depend on the small ...