Calculus of Variations and Geometric Measure Theory

G. Alberti - S. Baldo - G. Orlandi

Variational convergence for functionals of Ginzburg-Landau type

created on 23 Jan 2003
modified by alberti on 06 Mar 2007


Published Paper

Inserted: 23 jan 2003
Last Updated: 6 mar 2007

Journal: Indiana Univ. Math. J.
Volume: 54
Number: 5
Pages: 1411-1472
Year: 2005

Revised version, August 2003


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).

Keywords: minimal surfaces, Gamma convergence, integral currents, Ginzburg-Landau functionals, Plateau's problem