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

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