On the Grad–Rubin boundary value problem for the two-dimensional magneto-hydrostatic equations

In this work, we study the solvability of a boundary value problem for the magneto-hydrostatic equations originally proposed by Grad and Rubin (Proceedings of the 2nd UN conference on the peaceful uses of atomic energy. IAEA, Geneva, 1958). The proof relies on a fixed point argument which combines the so-called current transport method together with Hölder estimates for a class of non-convolution singular integral operators. The same method allows to solve an analogous boundary value problem for the steady incompressible Euler equations.