A variational interpretation of the Biot-Savart operator and the helicity of a bounded domain

In this note, we show that the projection of the Biot-Savart operator over the space of divergence-free vector fields that are tangential to the boundary is the solution of a suitable saddle-point variational problem. Since this projected Biot-Savart operator is shown to be compact, its spectrum can be completely characterized. In particular, through a suitable finite element discretization, it becomes possible to compute the helicity of a bounded domain of a general topological shape, via the determination of the eigenvalue of the projected Biot-Savart operator that has a maximum absolute value.