A free boundary problem for three-dimensional harmonic vector fields

We study in this paper a free boundary value problem ( FB ), where a region G_o in R ³ is determined by the condition that there exists a vector field v_o in G_o which satisfies div v_o = e_o, curl v_o = g_o in G_o and v_o = E on the boundary ?G_o with a given scalar function e_o and given vector fields g_o and E. We give two equivalent formulations for this problem. Then we characterize the solutions by a non-linear integral equation. In order to solve the latter by a Newton method we linearize this equation. We investigate the ensuing linear integral equation. In case of axisymmetric configurations this is a singular integral equation whose index can be easily determined from the given data. We obtain a related equation, if we try to construct a field v in a region G which is on the boundary perpendicular to a given field B . Finally we use this method to investigate an astrophysical problem, which arises in the theory of pulsar magnetospheres.