A Sharp Bound for the Growth of Minimal Graphs

Abstract

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 $$\begin{aligned} S_{\lambda }=\{(r,\theta )=\{-\lambda /2<\theta<\lambda /2\},\quad \pi <\lambda \le 2\pi , \end{aligned}$$then the rate of growth is at most \(r^{\pi /\lambda }\).