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arXiv · 1510.05288

Radial Limits of Bounded Nonparametric PMC Surfaces

Mozhgan Entekhabi·Kirk E. Lancaster
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Abstract

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

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Mozhgan Entekhabi, Kirk E. Lancaster. 2015-10-26. Radial Limits of Bounded Nonparametric PMC Surfaces. https://doi.org/10.2140/pjm.2016.283.341

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