Global uniform estimate for the modulus of $2D$ Ginzburg-Landau vortexless solutions with asymptotically infinite boundary energy

For $\epsilon >0$, let $u_{\epsilon }:\mathrm{\Omega }\to {\mathbb{R}}^2$ be a solution of the Ginzburg-Landau system $$-\mathrm{\Delta }{u}_{\epsilon }=\frac{1}{{\epsilon }^2}{u}_{\epsilon }(1-\mathrm{\mid }{u}_{\epsilon }{\mathrm{\mid }}^2)$$ in a Lipschitz bounded domain $\mathrm{\Omega }$. In an energy regime that excludes interior vortices, we prove that $1-\mathrm{\mid }{u}_{\epsilon }\mathrm{\mid }$ is uniformly estimated by a positive power of $\epsilon$ $globally$ in $\mathrm{\Omega }$ provided that the energy of $u_{\epsilon }$ at the boundary $\mathrm{\partial }\mathrm{\Omega }$ does not grow faster than ${\epsilon }^{-\alpha }$ with $\alpha \in (0,1)$.