Properties of the Biot–Savart Operator Acting on Surface Currents

SIAM Journal on Mathematical Analysis, Volume 56, Issue 5, Page 6446-6482, October 2024.
Abstract. We investigate properties of the image and kernel of the Biot–Savart operator in the context of stellarator designs for plasma fusion. We first show that for any given coil winding surface (CWS) the image of the Biot–Savart operator is [math]-dense in the space of square integrable harmonic fields defined on a plasma domain surrounded by the CWS. Then we show that harmonic fields which are harmonic in a proper neighborhood of the underlying plasma domain can in fact be approximated in any [math]-norm by elements of the image of the Biot–Savart operator. In the second part of this work we establish an explicit isomorphism between the space of harmonic Neumann fields and the kernel of the Biot–Savart operator which in particular implies that the dimension of the kernel of the Biot–Savart operator coincides with the genus of the CWS and hence turns out to be a homotopy invariant among regular domains in 3-space. Last, we provide an iterative scheme which we show converges weakly in [math]-topology to elements of the kernel of the Biot–Savart operator.