Radial limits of capillary surfaces at corners

Mozhgan (Nora) Entekhabi and Kirk E. Lancaster

Vol. 288 (2017), No. 1, 55–67

DOI: 10.2140/pjm.2017.288.55

Abstract

Consider a solution $f\in C^2\left(\Omega \right)$ of a prescribed mean curvature equation

$$div\frac{\nabla f}{\sqrt{1+|\nabla f{|}^2}}=2H\left(x,f\right)\text{ in }\Omega \subset {ℝ}^2,$$

where $\Omega$ is a domain whose boundary has a corner at $\mathcal{O}=\left(0,0\right)\in \partial \Omega$ and the angular measure of this corner is $2\alpha$, for some $\alpha \in \left(0,\pi \right)$. Suppose $\underset{x\in \Omega }{sup}\left|f\left(x\right)\right|$ and $\underset{x\in \Omega }{sup}\left|H\left(x,f\left(x\right)\right)\right|$ are both finite. If $\alpha >\frac{\pi }{2}$, then the (nontangential) radial limits of $f$ at $\mathcal{O}$, namely

$$Rf\left(\theta \right)=\underset{r\downarrow 0}{lim}f\left(rcos\theta ,rsin\theta \right),$$

were recently proven by the authors to exist, independent of the boundary behavior of $f$ on $\partial \Omega$, and to have a specific type of behavior.

Suppose $\alpha \in \left(\frac{\pi }{4},\frac{\pi }{2}\right]$, the contact angle $\gamma \left(\cdot \right)$ that the graph of $f$ makes with one side of $\partial \Omega$ has a limit (denoted ${\gamma }_2$) at $\mathcal{O}$ and

$$\pi -2\alpha <{\gamma }_2<2\alpha .$$

We prove that the (nontangential) radial limits of $f$ at $\mathcal{O}$ exist and the radial limits have a specific type of behavior, independent of the boundary behavior of $f$ on the other side of $\partial \Omega$. We also discuss the case $\alpha \in \left(0,\frac{\pi }{2}\right]$ and the displayed inequalities do not hold.

Keywords

prescribed mean curvature, radial limits

Mathematical Subject Classification 2010

Primary: 35B40, 53A10, 76D45, 35J93

Milestones

Received: 8 April 2016

Revised: 24 October 2016

Accepted: 30 October 2016

Published: 8 April 2017