J. Cantarella et al., Isoperimetric problems for the helicity of vector fields and the Biot-Savart and curl operators, J MATH PHYS, 41(8), 2000, pp. 5615-5641
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].