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

TL;DR

This paper establishes a global uniform estimate for the modulus of vortexless solutions to the 2D Ginzburg-Landau equations with boundary energy growing at a controlled rate, excluding interior vortices.

Contribution

It provides a new uniform estimate for the modulus of solutions in the vortexless regime with boundary energy growth constraints.

Findings

Uniform estimate of $1-|u_ ext{ε}|$ by a positive power of ε

Applicable to solutions with boundary energy growth up to ε^{- ext{α}}

Excludes interior vortices in the energy regime

Abstract

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 $\alpha\in (0,1)$.