A Sharp Bound for the Growth of Minimal Graphs

Allen Weitsman

doi:10.1007/s40315-021-00417-1

Weitsman, Allen

2021-12-01 00:00:00

We consider minimal graphs u=u(x,y)>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$u = u(x,y) > 0$$\end{document} over unbounded domains D⊂R2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D \subset R^2$$\end{document} bounded by a Jordan arc γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\gamma $$\end{document} on which u=0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$u = 0$$\end{document}. We prove a sort of reverse Phragmén-Lindelöf theorem by showing that if D contains a sector Sλ={(r,θ)={-λ/2<θ<λ/2},π<λ≤2π,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} S_{\lambda }=\{(r,\theta )=\{-\lambda /2<\theta<\lambda /2\},\quad \pi <\lambda \le 2\pi , \end{aligned}$$\end{document}then the rate of growth is at most rπ/λ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$r^{\pi /\lambda }$$\end{document}.

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Computational Methods and Function Theory Springer Journals

http://www.deepdyve.com/lp/springer-journals/a-sharp-bound-for-the-growth-of-minimal-graphs-0HQ2q3sNLB

$We consider minimal graphs u=u(x,y)>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$u = u(x,y) > 0$$\end{document} over unbounded domains D⊂R2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D \subset R^2$$\end{document} bounded by a Jordan arc γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\gamma $$\end{document} on which u=0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$u = 0$$\end{document}. We prove a sort of reverse Phragmén-Lindelöf theorem by showing that if D contains a sector Sλ={(r,θ)={-λ/2<θ<λ/2},π<λ≤2π,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} S_{\lambda }=\{(r,\theta )=\{-\lambda /2<\theta<\lambda /2\},\quad \pi <\lambda \le 2\pi , \end{aligned}$$\end{document}then the rate of growth is at most rπ/λ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$r^{\pi /\lambda }$$\end{document}.$

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A Sharp Bound for the Growth of Minimal Graphs

A Sharp Bound for the Growth of Minimal Graphs

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A Sharp Bound for the Growth of Minimal Graphs

A Sharp Bound for the Growth of Minimal Graphs

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