Radial limits of bounded nonparametric prescribed mean curvature surfaces

Mozhgan (Nora) Entekhabi and Kirk E. Lancaster

Vol. 283 (2016), No. 2, 341–351

DOI: 10.2140/pjm.2016.283.341

Abstract

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

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

where $\Omega \subset {ℝ}^2$ is a domain whose boundary has a corner at $\mathcal{O}=\left(0,0\right)\in \partial \Omega$. If $\underset{x\in \Omega }{sup}\left|f\left(x\right)\right|$ and $\underset{x\in \Omega }{sup}\left|H\left(x,f\left(x\right)\right)\right|$ are both finite and $\Omega$ has a reentrant corner at $\mathcal{O}$, then the (nontangential) radial limits of $f$ at $\mathcal{O}$,

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

are shown to exist, independent of the boundary behavior of $f$ on $\partial \Omega$, and to have a specific type of behavior. If $\underset{x\in \Omega }{sup}\left|f\left(x\right)\right|$ and $\underset{x\in \Omega }{sup}\left|H\left(x,f\left(x\right)\right)\right|$ are both finite and the trace of $f$ on one side has a limit at $\mathcal{O}$, then the (nontangential) radial limits of $f$ at $\mathcal{O}$ exist, the tangential radial limit of $f$ at $\mathcal{O}$ from one side exists and the radial limits have a specific type of behavior.

Dedicated to the memory of Alan Ross Elcrat

Keywords

prescribed mean curvature, radial limits

Mathematical Subject Classification 2010

Primary: 35B40, 53A10

Milestones

Received: 26 October 2015

Revised: 13 February 2016

Accepted: 15 February 2016

Published: 22 June 2016