A capillary surface with no radial limits

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. In Theorem 3, they provide an example of a capillary surface in a unit disk $D$ which has no radial limits at $(0,0)\in\partial D.$ 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 $(0,0)$ and find a capillary surface $z=f(x,y)$ in $\Omega\times\mathbb{R}$ which has no radial limits at $(0,0)\in\partial\Omega$ such that $\gamma$ is bounded away from $0$ and $\pi.$