#### Vol. 288, No. 1, 2017

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Radial limits of capillary surfaces at corners

### Mozhgan (Nora) Entekhabi and Kirk E. Lancaster

Vol. 288 (2017), No. 1, 55–67
##### Abstract

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

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

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

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 $\gamma \left(\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}\right)$ that the graph of $f$ makes with one side of $\partial \Omega$ has a limit (denoted ${\gamma }_{2}$) at $\mathsc{O}$ and

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

We prove that the (nontangential) radial limits of $f$ at $\mathsc{O}$ 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.