Radial Limits of Capillary Surfaces at Corners

Abstract

Consider a solution $f\in C^2(\Omega )$ 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 $\Omega$ is a domain whose boundary has a corner at $\mathcal{O}=(0,0)\in \partial \Omega$ and the angular measure of this corner is $2\alpha ,$ for some $\alpha \in (0,\pi ).$ Suppose ${\sup }_{x\in \Omega }|f(x)|$ and ${\sup }_{x\in \Omega }|H(x,f(x))|$ are both finite. If $\alpha >\frac{\pi }{2},$ then the (nontangential) radial limits of $f$ at $\mathcal{O},$ \[ Rf(θ) = \lim_{r\downarrow 0} f(r\cos(θ),r\sin(θ)), \] were recently proven by the authors to exist, independent of the boundary behavior of $f$ on $\partial \Omega ,$ and to have a specific type of behavior. Suppose $\alpha \in \left(\frac{\pi }{4},\frac{\pi }{2}\right],$ the contact angle $\gamma (\cdot )$ that the graph of $f$ makes with one side of $\partial \Omega$ has a limit (denoted ${\gamma }_2$) at $\mathcal{O}$ and \[ π-2α< γ_{2} <2α. \] 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 $\alpha \in \left(0,\frac{\pi }{2}\right].$

To be submitted to the Pacific Journal of Mathematics. arXiv admin note: text overlap with arXiv:1510.05288

References in corpus (1)

• Radial Limits of Bounded Nonparametric PMC Surfaces

Cited by in corpus (3)

• A Generalization of "Existence and Behavior of the Radial Limits of a Bounded Capillary Surface at a Corner"
• Radial Limits of Nonparametric PMC Surfaces with Intermediate Boundary Curvature
• Bernstein Functions and Radial Limits of Prescribed Mean Curvature Surfaces