Dust disks are discussed as models in astrophysics for certain galaxies and the matter in accretion disks around black holes. Since they are infinitesimally thin, the matter equations are ordinary differential equations whose solutions lead to boundary value problems for the Ernst equation. Since Riemann-Hilbert problems for the Ernst equation can be solved explicitly on Riemann surfaces, one can study which boundary value problems can be solved on a given Riemann surface. We restrict ourselves to hyperelliptic surfaces. On the axis there exists an algebraic relation between the real and the imaginary part of the Ernst equation. In the general case there exist algebraic relations between the Ernst potential and its derivatives. To establish these relations we use an algebraic representation of Korotkin's solutions. As an example we construct a class of solutions for counterrotating dust disks. We prove that the solutions are globally regular except at the disk up to the black hole limit. The complete metric is given in terms of hyperelliptic functions which are evaluated numerically. We discuss the black hole limit and the singular solutions in the region beyond this limit.