A criterion for constantness of harmonic functions on exterior domains
type: Item
instance of: scholarly article
title: A criterion for constantness of harmonic functions on exterior domains
zbMATH Open document ID: 1026.31002
DOI: 10.1016/S0096-3003(02)00357-0
published in: Applied Mathematics and Computation
publication date: 2003-07-30T00:00:00Z
review text: Let \(f: \mathbb{R}^2\to\mathbb{R}\) be a function that is bounded from below and satisfying the Laplace equation: \(f_{xx}+ f_{yy}= 0\). Then Liouville's theorem assures that \(f\) is constant. We can observe that, in general, the result is not true when the harmonic function \(f\) is defined on an exterior domain, that is the complement of a compact set. The main result of this paper is as follows. Let \(K\) be a non-empty convex set of \(\mathbb{R}^2\) and \(f: \mathbb{R}^2\setminus K\to\mathbb{R}\) a harmonic function, bounded from below. Assume that \[ \limsup_{\|(x,y)\|\to+\infty} {f(x,y)\over\|(x, y)\|}= 0. \] Then the function \(f\) is constant, if and only if the relation \[ |\nabla f(x,y)\cdot\nabla f_{xx}(x,y)|\leq \|\nabla f_x(x,y)\|^2, \] holds for every \((x,y)\in \mathbb{R}^2\setminus K\), where the symbol \(\cdot\) denotes the dot product of the gradients \(\nabla f\) and \(\nabla f_{xx}\).
reviewed by: Zbigniew Jerzy Jakubowski
Mathematics Subject Classification ID: 31A05
zbMATH DE Number: 1957190
zbMATH Keywords: harmonic functions on exterior domains
zbMATH Keywords: Liouville's theorem
author: Murat Ozdemir
MaRDI profile type: Publication
cites work: Q4881676
cites work: Q4317868
description: scientific article; zbMATH DE number 1957190