Global Minimality and Rigidity of the Constraint Map Vortex

Solution Description

We consider the minimization problem

\[ \min\left\{\int_{B_1}|Du|^2:\ u\in W^{1,2}(B_1;\mathbb R^n),\quad u=x\ \text{on }\partial B_1,\quad |u|\ge a\right\}, \quad 0< a< 1. \]

Figalli, Guerra, Kim, and Shahgholian proved that the canonical radial vortex is the unique global minimizer for $n\ge7$, and asked whether the same holds in dimensions \(3\le n \le6\). We answer this question affirmatively, thereby completing the global minimality and rigidity of the constraint map vortex in every dimension \(n\ge3\).

To be honest, we admit the use of AI tools (GPT 5.6 sol). We verified all mathematical statements and proofs independently, and we take full responsibility for the content of the manuscript.

1. Article Is an originGlobal Minimality and Rigidity of the Constraint Map Vortex

   • Bin D,
   • Jiahuan Li,
   • Yilu Liu,
   • Xi-Nan Ma

   arXiv