arXiv2016
Consider a solution $f\in C^{2}(Ω)$ of a prescribed mean curvature equation \[ {\rm div}\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^{2}}}\right)=2H(x,f) \ \ \ \ {\rm in} \ \ Ω\subset R^{2}, \] where $Ω$ is a domain whose boundary has a corner at ${\cal O}=(0,0)\in\partialΩ$ and the angular measure of this corner is $2α,$ for some $α\in (0,π).$ Suppose $\sup_{x\inΩ} |f(x)|$ and $\sup_{x\inΩ} |H(x,f(x))|$ are both finite. If $α>\fracπ{2},$ then the (nontangential) radial limits of $f$ at ${\cal O},$ \[ Rf(θ) = \lim_{r\downarrow 0} f(r\cos(θ),r\sin(θ)), \] were recently proven by the authors to exist, independent of the boundary behavior of $f$ on $\partialΩ,$ and to have a specific type of behavior. Suppose $α\in \left(\fracπ{4},\fracπ{2}\right],$ the contact angle $γ(\cdot)$ that the graph of $f$ makes with one side of $\partialΩ$ has a limit (denoted $γ_{2}$) at ${\cal O}$ and \[ π-2α< γ_{2} <2α. \] We prove that the (nontangential) radial limits of $f$ at ${\cal O}$ exist and the radial limits have a specific type of behavior, independent of the boundary behavior of $f$ on the other side of $\partialΩ.$ We also discuss the case $α\in \left(0,\fracπ{2}\right].$