C1-alpha Convergence of Minimizers of a Ginzburg-Landau Functional

Abstract

In this article we study the minimizers of the functional Eε(u, G) = 1/p ∫G | ∇u|p + 1/4εp ∫G (1 - |u|2)2, on the class Wg = {v ∈ W1,p (G, ℝ2); v|∂G = g}, where g : ∂G → S1 is a smooth map with Brouwer degree zero, and p is greater than 2. In particular, we show that the minimizer converges to the p-harmonic map in C1αloc (G, ℝ2) as ε approaches zero.