Radial Limits of Bounded Nonparametric PMC Surfaces
| Autor: | Entekhabi, Mozhgan, Lancaster, Kirk E. |
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| Rok vydání: | 2015 |
| Předmět: | |
| Zdroj: | Pacific J. Math. 283 (2016) 341-351 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.2140/pjm.2016.283.341 |
| Popis: | 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. Comment: 12 pages. Submitted to the Pacific Journal of Mathematics |
| Databáze: | arXiv |
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