Asymptotic behaviour of minimal graphs over exterior domains
type: Item
instance of: scholarly article
title: Asymptotic behaviour of minimal graphs over exterior domains
zbMATH Open document ID: 0632.49020
DOI: 10.1016/S0294-1449(16)30367-5
author: Leon Simon
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78130
review text: The author proves the following generalization of a well-known theorem by \textit{L. Bers} [Ann. Math., II. Ser. 53, 364-386 (1951; Zbl 0043.159)]. ``If u is a \(C^ 2\)-solution of the minimal surface equation over \({\mathbb{R}}^ n\sim \Omega\), \(\Omega\) bounded open, \(3\leq n\leq 7\), then Du(x) is bounded and has a limit as \(| x| \to \infty.''\) This follows from a more general theorem valid in any dimension \(n\geq 3\) concerning tangent cones of graph u at \(\infty\). To prove this general theorem one has to show that either Du(x) is bounded or all tangent cones of graph u at \(\infty\) are cylinders of the form \(C\times {\mathbb{R}}\) where \(C\subset {\mathbb{R}}^ n\) is an (n-1)-dimensional minimizing cone with \(\partial C=0\) and \(0\in \sin g C.\) Besides standard arguments from the regularity theory for codimension one minimal surfaces one has to use a special case of a general unique tangent cone result to show that sing \(C\neq \emptyset\).
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 4028258
zbMATH Keywords: minimal surface equation
zbMATH Keywords: tangent cones
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4028258