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

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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].

61Anzellotti, G., Baldo, S., and D. Percivale, Dimension reduction in variational problems, asymptotic development in rconvergence and thin structures in elasticity, Asymptotic Analysis 9 (1994) 61-100.We consider families of variational problems :F. over domains [}. whose extension in one or more directions is small compared to the extension in the other directions, and goes to zero while ~ tends to zero. We study then the "variational" convergence of the functionals:F. to a new functional defined on a domain A in a lower dimensional space, where those "dimensions" that were small in [}. disappear. A general framework is presented in the first part of the paper and an application to the elastic rod and the elastic plate is given in the second part.

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