Uniqueness of degree-one Ginzburg-Landau vortex in the unit ball in dimensions $N\ge 7$

Abstract

For $\epsilon >0$, we consider the Ginzburg-Landau functional for ${\mathbb{R}}^N$-valued maps defined in the unit ball $B^N\subset {\mathbb{R}}^N$ with the vortex boundary data $x$ on $\partial B^N$. In dimensions $N\ge 7$, we prove that for every $\epsilon >0$, there exists a unique global minimizer $u_{\epsilon }$ of this problem; moreover, $u_{\epsilon }$ is symmetric and of the form $u_{\epsilon }(x)=f_{\epsilon }(|x|)\frac{x}{|x|}$ for $x\in B^N$.