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arXiv · 2510.20533

Non-optimal domains for the helicity maximisation problem

Wadim Gerner
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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.

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Wadim Gerner. 2025-10-23. Non-optimal domains for the helicity maximisation problem. https://arxiv.org/abs/2510.20533

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