Isoperimetric problems for the helicity of vector fields and the Biot–Savart and curl operators

The helicity of a smooth vector field defined on a domain in three-space is the standard measure of the extent to which the field lines wrap and coil around one another. It plays important roles in fluid mechanics, magnetohydrodynamics, and plasma physics. The isoperimetric problem in this setting is to maximize helicity among all divergence-free vector fields of given energy, defined on and tangent to the boundary of all domains of given volume in three-space. The Biot–Savart operator starts with a divergence-free vector field defined on and tangent to the boundary of a domain in three-space, regards it as a distribution of electric current, and computes its magnetic field. Restricting the magnetic field to the given domain, we modify it by subtracting a gradient vector field so as to keep it divergence-free while making it tangent to the boundary of the domain. The resulting operator, when extended to the $L2$ completion of this family of vector fields, is compact and self-adjoint, and thus has a largest eigenvalue, whose corresponding eigenfields are smooth by elliptic regularity. The isoperimetric problem for this modified Biot–Savart operator is to maximize its largest eigenvalue among all domains of given volume in three-space. The curl operator, when restricted to the image of the modified Biot–Savart operator, is its inverse, and the isoperimetric problem for this restriction of the curl is to minimize its smallest positive eigenvalue among all domains of given volume in three-space. These three isoperimetric problems are equivalent to one another. In this paper, we will derive the first variation formulas appropriate to these problems, and use them to constrain the nature of any possible solution. For example, suppose that the vector field $V,$ defined on the compact, smoothly bounded domain Ω, maximizes helicity among all divergence-free vector fields of given nonzero energy, defined on and tangent to the boundary of all such domains of given volume. We will show that (1) $|V|$ is a nonzero constant on the boundary of each component of Ω; (2) all the components of ∂Ω are tori; and (3) the orbits of $V$ are geodesics on ∂Ω. Thus, among smooth simply connected domains, none are optimal in the above sense. In principal, one could have a smooth optimal domain in the shape, say, of a solid torus. However, we believe that there are no smooth optimal domains at all, regardless of topological type, and that the true optimizer looks like the singular domain presented in this paper, which we can think of either as an extreme apple, in which the north and south poles have been pressed together, or as an extreme solid torus, in which the hole has been shrunk to a point. A computational search for this singular optimal domain and the helicity-maximizing vector field on it is at present under way, guided by the first variation formulas in this paper.