Vol. 234, No. 2, 2008

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ISSN: 0030-8730
CMC capillary surfaces at reentrant corners

Maria Athanassenas and Kirk Lancaster

Vol. 234 (2008), No. 2, 201–228
Abstract

For a capillary graph in a vertical cylinder Ω × 3, the existence of a reentrant corner P Ω makes the determination of the continuity at P (or the behavior of the radial limits at P) of the solution problematic. Since continuity is the necessary consequence of the existence of a “central fan” of radial limits under certain conditions, the determination of necessary and sufficient conditions for the existence of a central fan is a very important open question in the mathematical theory of capillarity. Examples by Finn and Shi suggest that “central fans” may be very rare in the sense that arbitrarily small perturbations can eliminate them. In this note we obtain examples of capillary graphs (with zero mean curvature), each of which is continuous or has a central fan at a reentrant corner.

Keywords
minimal surface, capillary graph, Riemann–Hilbert problem
Mathematical Subject Classification 2000
Primary: 76B45
Secondary: 53A10, 76B03
Milestones
Received: 23 October 2006
Revised: 21 October 2007
Accepted: 12 November 2007
Published: 1 February 2008
Authors
Maria Athanassenas
School of Mathematical Sciences
PO Box 28M
Monash University
Clayton Campus VIC 3800
Australia
Kirk Lancaster
Department of Mathematics and Statistics
Wichita State University
Wichita, KA 67260-0033
United States