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

Abstract

We study critical points of the Ginzburg-Landau energy functional $$\mathcal{F}_\varepsilon[u] = \int_\Omega \Big[ \frac{1}{2}|\nabla u|^2 + \frac{1}{2\varepsilon^2} W(1 - |u|^2)\Big]\,dx, \quad u \in H^1(\Omega, \mathbb{R}^N),$$ with $\Omega \subset \mathbb{R}^M$, $\varepsilon>0$, $M,N \geq 2$ 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 $u_\varepsilon$ satisfies the global uniform bound $|u_\varepsilon| \leq 1$ in $\Omega$. Furthermore, if a family of critical points $(u_\varepsilon)$ converges (in energy) to a smooth $\mathbb{S}^{N-1}$-valued harmonic map in the limit $\varepsilon \to 0$, then we prove global uniform bounds for $(\Delta u_\varepsilon)_{\varepsilon>0}$ in $\Omega$ and, in particular, global H\"older convergence of the gradients $(\nabla u_\varepsilon)$ in $\Omega$ as $\varepsilon \to 0$.