Minimality of vortex solutions to Ginzburg--Landau type systems for gradient fields in the unit ball in dimension $N\ge 4$

Abstract

We prove that the degree-one vortex solution is the unique minimizer for the Ginzburg--Landau functional for gradient fields (that is, the Aviles--Giga model) in the unit ball $B^N$ in dimension $N\ge 4$ and with respect to its boundary value. A similar result is also proved for ${\mathbb{S}}^N$-valued maps in the theory of micromagnetics. Two methods are presented. The first method is an extension of the analogous technique previously used to treat the unconstrained Ginzburg--Landau functional in dimension $N\ge 7$. The second method uses a symmetrization procedure for gradient fields such that the $L^2$-norm is invariant while the $L^p$-norm, $2<p<\infty$, and the $H^1$-norm are lowered.