Radial Limits of Capillary Surfaces at Corners

Consider a solution $f\in C^2(\mathrm{\Omega })$ of a prescribed mean curvature equation $$\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{\mathrm{\nabla }f}{\sqrt{1+\mathrm{\mid }\mathrm{\nabla }f{\mathrm{\mid }}^2}}\right)=2H(x,f)\mathrm{i}\mathrm{n}\mathrm{\Omega }\subset R^2,$$ where $\mathrm{\Omega }$ is a domain whose boundary has a corner at ${\cal O}=(0,0)\in\partial\Omega$andtheangularmeasureofthiscorneris$2\alpha,$forsome$\alpha\in (0,\pi).$Suppose$\sup_{x\in\Omega} |f(x)|$and$\sup_{x\in\Omega} |H(x,f(x))|$arebothfinite\mathrm{.}If$\alpha>\frac{\pi}{2},$thenthe(nontangential)radiallimitsof$f$at${\cal O},$ $$Rf(\theta )=\underset{r\downarrow 0}{\lim }f(r\cos (\theta ),r\sin (\theta )),$$ were recently proven by the authors to exist, independent of the boundary behavior of $f$ on $\mathrm{\partial }\mathrm{\Omega },$ and to have a specific type of behavior. Suppose $\alpha \in (\frac{\pi }{4},\frac{\pi }{2}],$ the contact angle $\gamma (\cdot )$ that the graph of $f$ makes with one side of $\mathrm{\partial }\mathrm{\Omega }$ has a limit (denoted ${\gamma }_2$) at $\mathcal{O}$ and $$\pi -2\alpha <{\gamma }_2<2\alpha \mathrm{.}$$ 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 $\mathrm{\partial }\mathrm{\Omega }\mathrm{.}$ We also discuss the case $\alpha\in \left(0,\frac{\pi}{2}\right].$