We classify nonconstant entire local minimizers of the standard Ginzburg–Landau functional for maps in $H_{\text{loc}}^1({ℝ}^3;{ℝ}^3)$ satisfying a natural energy bound. Up to translations and rotations,such solutions of the Ginzburg–Landau system are given by an explicit solution equivariant under the action of the orthogonal group.

We consider the functional $${ℐ}_{\Omega }\left(v\right)={\int }_{\Omega }\left[f\right(\left|Dv\right|)-v]dx,$$ where $\Omega$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$, Crasta [Cr1] has shown that if ${ℐ}_{\Omega }$ admits a minimizer in $W_0^{1,1}\left(\Omega \right)$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball. With some restrictions on $f$, we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these...