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

• Verfasst von: Ripoll, Jaime [VerfasserIn]
• Verfasst von: Tomi, Friedrich [VerfasserIn]
• Titel: On solutions to the exterior Dirichlet problem for the minimal surface equation with catenoidal ends
• Verf.angabe: Jaime Ripoll, Friedrich Tomi
• Jahr: 2014
• Jahr des Originals: 2012
• Umfang: 22 S.
• Fussnoten: Online veröffentlicht: 15.12.2012 ; Gesehen am 29.09.2020
• Titel Quelle: Enthalten in: Advances in calculus of variations
• Ort Quelle: Berlin : de Gruyter, 2008
• Jahr Quelle: 2014
• Band/Heft Quelle: 7(2014), 2, Seite 205-226
• ISSN Quelle: 1864-8266
• DOI: doi:10.1515/acv-2012-0010
• Datenträger: Online-Ressource
• Sprache: eng
• K10plus-PPN: 1733915575

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.