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

Abstract

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

This new version contains improvements regarding the optimality of our assumptions, with explicit examples in section 4

Cited by in corpus (1)

• Global Jacobian and $\Gamma$-convergence in a two-dimensional Ginzburg-Landau model for boundary vortices