A sharp bound for the growth of minimal graphs. (English) Zbl 1491.53012
Summary: We consider minimal graphs \(u = u(x,y) > 0\) over unbounded domains \(D \subset R^2\) bounded by a Jordan arc \(\gamma\) on which \(u = 0\). We prove a sort of reverse Phragmén-Lindelöf theorem by showing that if \(D\) contains a sector
\[
S_{\lambda }=\{(r,\theta )=\{-\lambda /2<\theta<\lambda /2\},\quad \pi <\lambda \le 2\pi,
\]
then the rate of growth is at most \(r^{\pi /\lambda }\).
MSC:
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 35B53 | Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs |
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