Global minimality of the degree-one Ginzburg-Landau vortex solution in the unit ball in dimensions $n\ge 2$

Abstract

We consider the problem of minimizing the Ginzburg-Landau functional among ${\mathbb{R}}^n$-valued maps from the unit ball $B\subset {\mathbb{R}}^n$ with vortex boundary condition $u(x)=x$ on $\partial B$. We show that, for every Ginzburg-Landau parameter $\epsilon >0$, there exists a unique minimizer given exactly by the rotationally symmetric vortex solution in dimension $n\ge 2$.