ON CONSTANT MEAN CURVATURE GRAPHS WITH CONVEX BOUNDARY

Title & Authors
ON CONSTANT MEAN CURVATURE GRAPHS WITH CONVEX BOUNDARY
Park, Sung-Ho;

Abstract
We give area and height estimates for cmc-graphs over a bounded planar $\small{C^{2,{\alpha}}}$ domain $\small{{\Omega}{\subset}\mathbb{R}^3}$. For a constant H satisfying $\small{H^2{\mid}{\Omega}{\mid}{\leq}9{\pi}/16}$, we show that the height $\small{h}$ of H-graphs over $\small{{\Omega}}$ with vanishing boundary satisfies $\small{{\mid}h{\mid}}$ < $\small{(\tilde{r}/2{\pi})H{\mid}{\Omega}{\mid}}$, where $\small{\tilde{r}}$ is the middle zero of $\small{(x-1)(H^2{\mid}{\Omega}{\mid}(x+2)^2-9{\pi}(x-1))}$. We use this height estimate to prove the following existence result for cmc H-graphs: for a constant H satisfying $\small{H^2{\mid}{\Omega}{\mid}}$ < $\small{(\sqrt{297}-13){\pi}/8}$, there exists an H-graph with vanishing boundary.
Keywords
constant mean curvature;height estimate;Dirichlet problem;
Language
English
Cited by
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