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Global minimality and rigidity of the constraint map vortex

Let 3≤n≤63\le n\le 6 and 0<a<10<a<1. Consider the variational problem min⁡{∫B1∣Du∣2 dx: u∈W1,2(B1;Rn), u=x on ∂B1, ∣u∣≥a a.e. in B1}\min\left\{\int_{B_1}|Du|^2\,dx:\ u\in W^{1,2}(B_1;\mathbb{R}^n),\ u=x\ \text{on }\partial B_1,\ |u|\ge a\ \text{a.e. in }B_1\right\}. Is the canonical radial constraint-map vortex the unique global minimizer?

References

Primary source

arXiv

Additional references

calculus-of-variationsgeometric-analysispartial-differential-equations
Posted

Progress summary

Refreshed
Claimed solved

A new preprint claims to settle the remaining dimensions, making the canonical vortex the unique global minimizer throughout the problem’s stated range, but the proof has not been independently confirmed.

The problem asks whether the canonical, radial constraint-map vortex is globally minimizing and rigid in the dimensions left open by Figalli, Guerra, Kim, and Shahgholian. Before the latest development, minimality was known in dimensions n≥7n \ge 7 while the low-dimensional cases remained unresolved.

August 25, 2026 preprint

A new preprint claims to extend the theorem from n≥7n \ge 7 to 3≤n≤63 \le n \le 6, proving global minimality and uniqueness of the canonical minimizer in all dimensions covered by the problem. This is a claimed complete resolution, but the supplied evidence is only a preprint and contains no independent verification.

Current status (as of August 2026): Minimality is known for n≥7n \ge 7 and is claimed for 3≤n≤63 \le n \le 6 by the new preprint; the claimed completion remains unverified.

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