Global Minimality and Rigidity of the Constraint Map Vortex

We consider the minimization problem $$\min \{{\int }_{B_1}\mathrm{\mid }Du{\mathrm{\mid }}^2:u\in W^{1,2}(B_1;{\mathbb{R}}^n),u=x\text{ on }\mathrm{\partial }{B}_1,\mathrm{\mid }u\mathrm{\mid }\ge a\},0<a<1.$$ Figalli, Guerra, Kim, and Shahgholian proved that the canonical radial vortex is the unique global minimizer for $n\ge 7$, and asked whether the same holds in dimensions $3\le n\le 6$. We answer this question affirmatively, thereby completing the global minimality and rigidity of the constraint map vortex in every dimension $n\ge 3$.