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

We study critical points of the Ginzburg-Landau energy functional $${\mathcal{F}}_{\epsilon }[u]={\int }_{\mathrm{\Omega }}[\frac{1}{2}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2+\frac{1}{2{\epsilon }^2}W(1-\mathrm{\mid }u{\mathrm{\mid }}^2)]dx,u\in H^1(\mathrm{\Omega },{\mathbb{R}}^N),$$ with $\mathrm{\Omega }\subset {\mathbb{R}}^M$, $\epsilon >0$, $M,N\ge 2$ 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 $\mathrm{\partial }\mathrm{\Omega }$, we prove the following maximum principle: every critical point $u_{\epsilon }$ satisfies the global uniform bound $\mathrm{\mid }{u}_{\epsilon }\mathrm{\mid }\le 1$ in $\mathrm{\Omega }$. 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 $(\mathrm{\Delta }{u}_{\epsilon }{)}_{\epsilon >0}$ in $\mathrm{\Omega }$ and, in particular, global Hölder convergence of the gradients $(\mathrm{\nabla }{u}_{\epsilon })$ in $\mathrm{\Omega }$ as $\epsilon \to 0$.