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

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, $2<p<\mathrm{\infty }$, and the $H^1$-norm are lowered.