In the first part of this paper we prove that functionals of Ginzburg-Landau type for maps from a domain in dimension n+k into R^k converge in a suitable sense to the area functional for surfaces of dimension n (Theorem 1.1). In the second part we modify this result in order to include Dirichlet boundary condition (Theorem 5.5), and, as a corollary, we show that the rescaled energy densities and the Jacobians of minimizers converge to minimal surfaces of dimension n (Corollaries 1.2 and 5.6). Some of these results were announced in the paper "Un risultato di convergenza variazionale per funzionali di tipo Ginzburg-Landau in dimensione qualunque" by the first author

If $u$ is a function of bounded variation from the open set $\Omega \subset R^n$ into $R^m$, then $Du$ is a measure on $\Omega$ which takes values in the space of $m\times n$ matrices, and we denote by $D_Su$ the singular part of this measure (with respect to Lebesgue measure). We prove that that the density of $D_su$ with respect to its variation $|D_su|$ is a function with values in rank-one matrices. More generally, we show that given a singular measure $\mu$, there exists a unit vectorfield $\nu$ such that for every scalar $BV$ function $u$, the density of $Du$ with respect to $\mu$ at $x$ is a multiple of $\nu(x)$ for $\mu$-almost every $x$

We study minimizers of a nonlocal variational problem. The problem is a mathematical paradigm for the ubiquitous phenomenon of energy-driven pattern formation induced by competing short- and long-range interactions. The short-range interaction is attractive and comes from an interfacial energy, and the long-range interaction is repulsive and comes from a nonlocal energy contribution. In particular, the problem is the sharp interface version of a problem used to model microphase separation of diblock copolymers. A natural conjecture is that in all space dimensions, minimizers are essentially periodic on an intrinsic scale. However, proving any periodicity result turns out to be a formidable task in dimensions larger than one. In this paper, we address a weaker statement concerning the distribution of energy for minimizers. We prove in any space dimension that each component of the energy (interfacial and nonlocal) of any minimizer is uniformly distributed on cubes which are sufficiently large with respect to the intrinsic length scale

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The paper is concerned with the fine properties of monotone functions on R n . We study the continuity and differentiability properties of these functions, the approximability properties, the structure of the distributional derivatives and of the weak Jacobians. Moreover, we exhibit an example of a monotone function u which is the gradient of a C 1,α convex function and whose weak Jacobian Ju is supported on a purely unrectifiable set.

We present a minimality criterion for the Mumford-Shah functional, and more generally for non convex variational integrals on SBV which couple a surface and a bulk term. This method provides short and easy proofs for several minimality results. (2000): 49K10 (49Q15, 49Q05, 58E12). Mathematics Subject Classification

In this paper we consider a non-local anisotropic model for phase separation in two-phase fluids at equilibrium, and show that when the thickness of the interface tends to zero in a suitable way, the classical surface tension model is recovered. Relevant examples are given by continuum limits of ferromagnetic Ising systems in equilibrium statistical mechanics.

Abstract. 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 k−1 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 of codimension k can be realized as Jacobian of a Sobolev map valued in S k−1 . In case M is polyhedral, the map we construct is smooth outside M plus an additional polyhedral set of lower dimension, and can be used in the constructive part of the proof of a -convergence result for functionals of Ginzburg-Landau type, as described in [2].