Dirichlet boundary value problems of the Ernst equation

We demonstrate how the solution to an exterior Dirichlet boundary value problem of the axisymmetric, stationary Einstein equations can be found in terms of generalized solutions of the B\"acklund type. The proof that this generalization procedure is valid is given, which also proves conjectures about earlier representations of the gravitational field corresponding to rotating disks of dust in terms of B\"acklund-type solutions. As a further result, we find that, in contrast with the Laplace equation, arbitrary boundary values may not be prescribed.