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

TL;DR

This paper proves the uniqueness and symmetry of the degree-one Ginzburg-Landau vortex minimizer in the unit ball for dimensions N ≥ 7, for all positive epsilon, highlighting a special structure in high dimensions.

Contribution

It establishes the existence, uniqueness, and explicit symmetric form of the global minimizer for the Ginzburg-Landau functional in high dimensions, extending understanding of vortex solutions.

Findings

Unique global minimizer exists for all epsilon>0 in N≥7 dimensions.

Minimizer is symmetric and radially structured.

Explicit form of the minimizer as a radial profile times a unit vector.

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 $N\geq 7$, 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 $x\in B^N$.