On the Convergence of Solutions for the Ginzburg-Landau Equation and System

Abstract

Let $(u_{\epsilon })$ be a family of solutions of the Ginzburg--Landau equation with boundary condition $u_{\epsilon }=g$ on $\partial \Omega$ and of degree $0$. Let $u_0$ denote the harmonic map satisfying $u_0=g$ on $\partial \Omega$. We show that, if there exists a constant $C_1>0$ such that for $\epsilon$ sufficiently small we have $\frac{1}{2} \int_Ω|\nabla u_\ve|^2 dx \leq C_1 \leq \frac{1}{2} \int_Ω|\nabla u_0|^2 dx,$ then $C_1=\frac{1}{2}{\int }_{\Omega }|\nabla u_0{|}^{2}dx$ and $u_\ve ~\to ~ u_0 \qin H^1(\Om)$. We also prove that if there is a constant $C_2$ such that for $\ve$ small enough we have $ \frac12 \int_\Om |\nabla u_\ve|^2 dx \geq C_2 > \frac12 \int_\Om |\nabla u_0|^2 dx,$ then $|u_{\ve}|$ does not converge uniformly to $1$ on $\overline{\Om} $. We obtain analogous results for both symmetric and non-symmetric two-component Ginzburg--Landau systems.