On solutions to the exterior Dirichlet problem for the minimal surface equation with catenoidal ends - Ripoll, Jaime; Tomi, Friedrich (Advances in calculus of variations, 2014)
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.
Online veröffentlicht: 15.12.2012
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