Variational convergence for functionals of Ginzburg-Landau type

We consider functionals of Ginzburg-Landau type for maps defined on (n+k)-dimensional domain with values in the k-dimensional Euclidean space. In the first part of the paper we prove that these functionals converge in a suitable sense to the area functional for surfaces (more precisely, integral currents) of dimension n (Theorem 1.1). In the second part we modify this result in order to include Dirichlet boundary condition in suitable trace spaces (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).