CMC capillary surfaces at reentrant corners

Abstract

For a capillary graph in a vertical cylinder × ‫ޒ‬ ⊂ ‫ޒ‬ 3 , the existence of a reentrant corner P ∈ ∂ 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.

Untitled

“…This is a Riemann-Hilbert problem with discontinuous coefficients G; in the notation of [Monakhov 1983, Chapter 1, §4], this is a "Hilbert problem with piecewise Hölder coefficients" (see also [Athanassenas and Lancaster 2004]). In order to use the results in [Monakhov 1983], we need to compute the index of this Hilbert problem in an appropriate function class O(m) = O(t k 1 , .…”