Non-optimal domains for the helicity maximisation problem

TL;DR

This paper investigates the problem of maximizing Biot-Savart helicity in fixed-volume domains, establishing geometric constraints that exclude many solid tori from being optimal, while leaving the existence of optimal domains unresolved.

Contribution

It extends previous results by deriving new geometric conditions that optimal domains must satisfy, ruling out many candidates and highlighting open questions.

Findings

Certain solid tori cannot be optimal domains due to geometric constraints.

The existence of an optimal domain for the helicity maximization problem remains unresolved.

Abstract

In [J. Cantarella, D. DeTurck, H. Gluck and M. Teytel, J. Math. Phys. 41:5615 (2000)] the helicity isoperimetric problem which asks to find a smooth domain of fixed volume which maximises Biot-Savart helicity among all other smooth domains of fixed volume was initiated. It was shown that if an optimal domain exists, all of its boundary components must be tori. The present work extends these results by establishing additional geometric constraints which optimal domains, if they exist, must satisfy. This allows to rule out the optimality of a broad class of solid tori. The existence of optimal domains remains an open problem.