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

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 pro...

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Hauptverfasser: Ripoll, Jaime (VerfasserIn) , Tomi, Friedrich (VerfasserIn)
Dokumenttyp: Article (Journal)
Sprache:Englisch
Veröffentlicht: 2014
In: Advances in calculus of variations
Year: 2012, Jahrgang: 7, Heft: 2, Pages: 205-226
ISSN:1864-8266
DOI:10.1515/acv-2012-0010
Online-Zugang:Resolving-System, lizenzpflichtig, Volltext: https://doi.org/10.1515/acv-2012-0010
Verlag, lizenzpflichtig, Volltext: https://www.degruyterbrill.com/view/journals/acv/7/2/article-p205.xml
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Verfasserangaben:Jaime Ripoll, Friedrich Tomi

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520 |a 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. 
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