A capillary surface with no radial limits

Colm Patric Mitchell

Vol. 293 (2018), No. 1, 173–178

DOI: 10.2140/pjm.2018.293.173

Abstract

In 1996, Kirk Lancaster and David Siegel investigated the existence and behavior of radial limits at a corner of the boundary of the domain of solutions of capillary and other prescribed mean curvature problems with contact angle boundary data. They provided an example of a capillary surface in a unit disk $D$ which has no radial limits at $\left(0,0\right)\in \partial D$. In their example, the contact angle, $\gamma$, cannot be bounded away from zero and $\pi$. Here we consider a domain $\Omega$ with a convex corner at $\left(0,0\right)$ and find a capillary surface $z=f\left(x,y\right)$ in $\Omega \times ℝ$ which has no radial limits at $\left(0,0\right)\in \partial \Omega$ such that $\gamma$ is bounded away from $0$ and $\pi$.

Keywords

capillary surfaces, PDE, Concus–Finn conjecture

Mathematical Subject Classification 2010

Primary: 35B40, 35J93, 53A10

Milestones

Received: 1 February 2017

Accepted: 28 August 2017

Published: 3 November 2017