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, magnetohyd rodynamics, and plasma physics. The isoperimetric problem in this setting i s to maximize helicity among all divergence-free vector fields of given ene rgy, 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 vect or 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 magneti c field. Restricting the magnetic field to the given domain, we modify it b y subtracting a gradient vector field so as to keep it divergence-free whil e making it tangent to the boundary of the domain. The resulting operator, when extended to the L-2 completion of this family of vector fields, is com pact and self-adjoint, and thus has a largest eigenvalue, whose correspondi ng eigenfields are smooth by elliptic regularity. The isoperimetric problem for this modified Biot-Savart operator is to maximize its largest eigenval ue among all domains of given volume in three-space. The curl operator, whe n restricted to the image of the modified Biot-Savart operator, is its inve rse, and the isoperimetric problem for this restriction of the curl is to m inimize its smallest positive eigenvalue among all domains of given volume in three-space. These three isoperimetric problems are equivalent to one an other. In this paper, we will derive the first variation formulas appropria te 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 comp act, smoothly bounded domain Omega, 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 Omega; (2) all the components of partial derivative Omega are tori; and (3) the orbits of V ar e geodesics on partial derivative Omega. Thus, among smooth simply connecte d domains, none are optimal in the above sense. In principal, one could hav e 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 topo logical type, and that the true optimizer looks like the singular domain pr esented 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 extre me solid torus, in which the hole has been shrunk to a point. A computation al search for this singular optimal domain and the helicity-maximizing vect or field on it is at present under way, guided by the first variation formu las in this paper. (C) 2000 American Institute of Physics. [S0022-2488(00)0 0705-2].