The Ginzburg-Landau system with general potential: maximum principle and gradient estimates

Abstract

We study critical points of the Ginzburg-Landau energy functional $${\mathcal{F}}_{\epsilon }[u]={\int }_{\Omega }[\frac{1}{2}|\nabla u{|}^2+\frac{1}{2{\epsilon }^2}W(1-|u{|}^2)]dx,u\in H^1(\Omega ,{\mathbb{R}}^N),$$ with $\Omega \subset {\mathbb{R}}^M$, $\epsilon >0$, $M,N\ge 2$ and general conditions on the non-negative potential $W$ allowing for super-quadratic behaviour near its zero set. Under a Dirichlet boundary data of unit-length on $\partial \Omega$, we prove the following maximum principle: every critical point $u_{\epsilon }$ satisfies the global uniform bound $|u_{\epsilon }|\le 1$ in $\Omega$. Furthermore, if a family of critical points $(u_{\epsilon })$ converges (in energy) to a smooth ${\mathbb{S}}^{N-1}$-valued harmonic map in the limit $\epsilon \to 0$, then we prove global uniform bounds for $(\Delta u_{\epsilon }{)}_{\epsilon >0}$ in $\Omega$ and, in particular, global Hölder convergence of the gradients $(\nabla u_{\epsilon })$ in $\Omega$ as $\epsilon \to 0$.