Global continuity estimates for two dimensional graphs of prescribed Gauss curvature

Research Report MRR00-030

Global continuity estimates for two dimensional graphs of prescribed Gauss curvature

John Urbas

Abstract: We prove the global Hölder continuity of convex solutions $u\in C^3(\Omega)$ of the equation of prescribed positive Gauss curvature in a bounded convex domain $\Omega \subset \mathbf{R}^2$ with $\partial\Omega \in C^{1,\beta}$ for some $\beta\in (0,1]$. We also obtain better regularity for the trace of $u$ on $\partial\Omega$. In the special case $\beta=1$ we show that $u\in C^{0,1/2}(\bar{\Omega})$ and $u|_{\partial\Omega}\in C^{0,2/3}(\partial\Omega)$. We also investigate the global continuity of solutions in $C^1$ domains and construct an example showing that global continuity need not hold in general convex domains.

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