article

Title: Convergence of minimizers with local energy bounds for the Ginzburg-Landau functionals

Authors: S. Baldo, G. Orlandi and S. Weitkamp

Issue: Volume 58 (2009), Issue 5, 2369-2408

Abstract:

$We study the asymptotic behaviour, as $\epsilon \to 0$, of a sequence $\{u_{\epsilon}\}$ of minimizers for the Ginzburg-Landau functional which satisfies local energy bounds of order $|\log \epsilon|$. The jacobians $Ju_{\epsilon}$ are shown to converge, in a suitable sense and up to subsequences, to an area minimizing minimal surface of codimension $2$. This is achieved without assumptions on the global energy of the sequence or on the boundary data, and holds even for unbounded domains. The proof is based on an improved version of the $\Gamma$-convergence results from [G. Alberti, S. Baldo, and G. Orlandi, \emph{Variational convergence for functionals of Ginzburg-Landau type}, Indiana Univ. Math. J. \textbf{54} (2005 (5)), 1411--1472}.$