Radial Limits of Bounded Nonparametric PMC Surfaces

TL;DR

This paper investigates the behavior of solutions to a prescribed mean curvature equation near reentrant corners, establishing the existence and specific nature of radial limits of the solution at such boundary points.

Contribution

It proves the existence and characterizes the behavior of radial limits of bounded solutions to the prescribed mean curvature equation at reentrant corners, regardless of boundary data.

Findings

Radial limits of solutions exist at reentrant corners.

Radial limits exhibit a specific, predictable behavior.

Results are independent of boundary behavior on the domain boundary.

Abstract

Consider a solution $f\in C^{2}(\Omega)$ of a prescribed mean curvature equation \[ {\rm div}\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^{2}}}\right)=2H(x,f) \ \ \ \ {\rm in} \ \ \Omega, \] where $\Omega\subset \Real^{2}$ is a domain whose boundary has a corner at ${\cal O}=(0,0)\in\partial\Omega.$ 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 ${\cal O},$ then the radial limits of $f$ at ${\cal O},$ \[ Rf(\theta) \myeq \lim_{r\downarrow 0} f(r\cos(\theta),r\sin(\theta)), \] 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 ${\cal O},$ then the radial limits of $f$ at ${\cal O}$ exist and have a specific type of behavior.