Consider a solution
$f\in {C}^{2}\left(\Omega \right)$
of a prescribed mean curvature equation
$$div\frac{\nabla f}{\phantom{\rule{0.3em}{0ex}}\sqrt{1+\nabla f{}^{2}}}=2H\left(x,f\right)\phantom{\rule{1em}{0ex}}\text{in}\Omega \subset {\mathbb{R}}^{2},$$
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
$\underset{x\in \Omega}{sup}\leftf\left(x\right)\right$ and
$\underset{x\in \Omega}{sup}\leftH\left(x,f\left(x\right)\right)\right$ 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
$\gamma \left(\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}\right)$ that
the graph of
$f$ makes
with one side of
$\partial \Omega $ has
a limit (denoted
${\gamma}_{2}$)
at
$\mathcal{O}$
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.
