Asymptotic values of minimal graphs in a disc
type: Item
instance of: scholarly article
title: Asymptotic values of minimal graphs in a disc
zbMATH Open document ID: 1239.53004
author: Pascal Collin
author: Harold Rosenberg
publication date: 2011-10-06T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116337
review text: Let \(D\) be the open unit disk \(\{ (r,\theta) : 0\leq r<1,\;0\leq \theta\leq 2\pi\}\). For certain functions solving variational equations on \(D\), the authors investigate several questions regarding the existence of radial limits (in the extended reals) at the boundary of \(D\). In the first instance, the authors use the Poincaré disk model of hyperbolic space. They construct an unbounded minimal graph over the disk which fails to have a radial limit (in the extended reals) for almost every direction \(\theta\). In the second instance, the authors prove that a solution \(u\) of the prescribed mean curvature equation (over the Euclidean unit disk) \[ \text{div} \left( \frac{\nabla u}{\sqrt{1 + |Du|^2}}\right) = 2H \,, \] with \(H\) bounded and continuous, does have a radial limit (in the extended reals) for almost every direction \(\theta\). The proof is succinct and self-contained. In the third instance, the authors construct an example of a minimal graph over the Euclidean unit disk for which there is but a measure zero set of directions \(\theta\) for which there exists a finite radial limit (in the extended reals). This construction is based on extending Scherk surfaces and is inspired by the example in the authors' paper [Ann. Math. (2) 172, No. 3, 1879--1906 (2010; Zbl 1209.53010)].
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C43
zbMATH DE Number: 5954673
zbMATH Keywords: minimal graphs
zbMATH Keywords: radial limits
zbMATH Keywords: Fatou theorem
reviewed by: Harold Parks
MaRDI profile type: Publication
OpenAlex ID: W2322395326
cites work: Construction of harmonic diffeomorphisms and minimal graphs
cites work: Variational problems of minimal surface type. II: Boundary value problems for the minimal surface equation
cites work: Two theorems on boundary properties of minimal surfaces in nonparametric form
cites work: On new results in the theory of minimal surfaces
DOI: 10.5802/AIF.2610
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5954673