CMC capillary surfaces at reentrant corners

Maria Athanassenas and Kirk Lancaster

Vol. 234 (2008), No. 2, 201–228

DOI: 10.2140/pjm.2008.234.201

Abstract

For a capillary graph in a vertical cylinder Ω × ℝ ⊂ ℝ³, 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