Non-optimal domains for the helicity maximisation problem

2025/10/23 by Wadim Gerner, Gerner, Wadim
Mathematics · #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows #Analytic and geometric function theory

paper · pdf · doi:10.48550/arxiv.2510.20533

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.

Citations

• Asymptotic windings, surface helicity and their applications in plasma physics
• Existence of optimal domains for the helicity maximisation problem among domains satisfying a uniform ball condition
• Optimal convex domains for the first curl eigenvalue
• Non-existence of axisymmetric optimal domains with smooth boundary for the first curl eigenvalue

Related

• Existence of optimal domains for the helicity maximisation problem among domains satisfying a uniform ball condition
• Non-existence of axisymmetric optimal domains with smooth boundary for the first curl eigenvalue
• Existence of an optimal domain for minimizing the fundamental tone of a clamped plate of prescribed volume in arbitrary dimension
• A shape optimization problem in cylinders and related overdetermined problems
• Shape optimization of harmonic helicity in toroidal domains