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In this paper we investigate the Dirichlet problem for the minimal surface equation on certain nonconvex domains of the plane. In our first result, we give, by an independent proof, a numerically explicit version of Williams' existence theorem. Our main result concerns the Dirichlet problem on exterior domains. It was shown by Krust (1989) and Kuwert (1993) that between two different solutions with the same normal at infinity there is a continuum of solutions foliating the space in between. We investigate the space of solutions further and show that, unless it is empty, it contains a maximal and a minimal solution if the boundary data is rectifiable. In the case of sufficiently smooth data we parametrize the set of solutions in terms of the extremal inclinations which the normal of the graph of a solution reaches at the boundary. We show that all theoretically possible values are realized including the horizontal position of the normal for the minimal and maximal solutions. We moreover give an example where the maximal and the minimal solution coincide so that there is exactly one with given normal at infinity. This answers a natural question which has not been touched in the previous papers.

On solutions to the exterior Dirichlet problem for the minimal surface equation with catenoidal ends