Radial Limits of Bounded Nonparametric PMC Surfaces

Abstract

Consider a solution $f\in C^2(\Omega )$ of a prescribed mean curvature equation \[ {\rm div}\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^{2}}}\right)=2H(x,f) \ \ \ \ {\rm in} \ \ Ω, \] where $Ω\subset \Real^{2}$ is a domain whose boundary has a corner at $\mathcal{O}=(0,0)\in \partial \Omega .$ If ${\sup }_{x\in \Omega }|f(x)|$ and ${\sup }_{x\in \Omega }|H(x,f(x))|$ are both finite and $\Omega$ has a reentrant corner at $\mathcal{O},$ then the radial limits of $f$ at $\mathcal{O},$ \[ Rf(θ) \myeq \lim_{r\downarrow 0} f(r\cos(θ),r\sin(θ)), \] are shown to exist and to have a specific type of behavior, independent of the boundary behavior of $f$ on $\partial \Omega .$ If ${\sup }_{x\in \Omega }|f(x)|$ and ${\sup }_{x\in \Omega }|H(x,f(x))|$ are both finite and the trace of $f$ on one side has a limit at $\mathcal{O},$ then the radial limits of $f$ at $\mathcal{O}$ exist and have a specific type of behavior.

12 pages. Submitted to the Pacific Journal of Mathematics

Cited by in corpus (2)

• Radial Limits of Capillary Surfaces at Corners
• A Generalization of "Existence and Behavior of the Radial Limits of a Bounded Capillary Surface at a Corner"