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We prove that the degree-one vortex solution is the unique minimizer for theGinzburg--Landau functional for gradient fields (that is, the Aviles--Gigamodel) in the unit ball $B^N$ in dimension $N \geq 4$ and with respect to itsboundary value. A similar result is also proved for $\mathbb{S}^N$-valued mapsin the theory of micromagnetics. Two methods are presented. The first method isan extension of the analogous technique previously used to treat theunconstrained Ginzburg--Landau functional in dimension $N \geq 7$. The secondmethod 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.

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