A Liouville-type theorem for harmonic functions on exterior domains
type: Item
instance of: scholarly article
title: A Liouville-type theorem for harmonic functions on exterior domains
zbMATH Open document ID: 0956.31001
author: Antonia Chinnì
author: F. Cammaroto
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-03-12T00:00:00Z
review text: Liouville's classical theorem that a function harmonic in \(\mathbb{R}^2\) that is bounded below is a constant does not extend to \(\mathbb{R}^2\) punctured at the origin. Via some interesting preliminaries on convex sets in \(\mathbb{R}^2\), the authors prove that if \(K\) is a non-empty compact convex set in \(\mathbb{R}^2\) and \(f\) is a real harmonic function in \(\mathbb{R}^2\setminus K\) that is bounded below and satisfies the condition that \(f(x,y)= o(\|(x,y)\|)\) as \((x,y)\to \infty\), then \(f\) is constant if \[ \bigl|\nabla f(x,y) \bigr|\cdot\bigl|\nabla f_{x,x}(x,y) \bigr|\leq \bigl|\nabla f_x(x,y) \bigr|^2 \] for every \((x,y)\) in \(\mathbb{R}^2\setminus K\).
Mathematics Subject Classification ID: 31A05
zbMATH DE Number: 1441382
zbMATH Keywords: harmonic functions
zbMATH Keywords: Liouville-type theorem
reviewed by: David A. Brannan
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6660
OpenAlex ID: W2069833927
cites work: Q4881676
cites work: Q4317868
DOI: 10.1006/JMAA.1999.6660
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description: scientific article; zbMATH DE number 1441382