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

Abstract

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.

The Dirichlet problem for the minimal hypersurface equation on arbitrary domains of a Riemannian manifold

“…Williams' results (existence and non existence) were extended to unbounded domain Ω ⊂ R 2 by N. Kutev and F. Tomi in [8] and J. Ripoll and F. Tomi gave, in the specific case Ω ⊂ R 2 , Williams' condition in a more explicit form (see Theorem 1 of [11]). Schulz and Williams [12] and Bergner [2] generalized Williams' result to prescribed mean curvature (in Euclidean spaces).…”

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“…• Unbounded domains of ‫ޒ‬ 2 : [Hwang 1988;Collin and Krust 1991;Sá Earp and Rosenberg 1989;Ripoll and Tomi 2014;Krust 1989;Kuwert 1993;Kutev and Tomi 1998].…”

Asymptotic and exterior Dirichlet problems for the minimal surface equation in the Heisenberg group with a balanced metric

“…R. Krust proved that the solutions of the EDP having the same Gauss map at infinity form a foliation if there are at least two solutions ( [7]). Krust's result was improved by J. Ripoll and F. Tomi in [17] where they proved the existence of a minimal and of a maximal solutions and also the existence of a boundary data admitting exactly one solution. The results of Krust were extended to arbitrary dimensions by E. Kuwert in [9].…”