On p-harmonic functions in the plane and their stream functions
type: Item
instance of: scholarly article
title: On p-harmonic functions in the plane and their stream functions
zbMATH Open document ID: 0659.35037
DOI: 10.1016/0022-0396(88)90022-8
author: Gunnar Aronsson
author: Peter Lindqvist
published in: Journal of Differential Equations
publication date: 1988-01-01T00:00:00Z
review text: Suppose that D is a domain in \({\mathbb{R}}^ 2\) and \(p>1\). A function u is called p-harmonic in D if u is in the Sobolev space \(W^{1,p}_{loc}(D)\) and satisfies \(div(| \nabla u|^{p-2} \nabla u)=0\). The paper presents itself as a survey article but also contains new results. For instance it is proved that if \(\phi\) is p- harmonic in D but is not a polynomial of degree 1, then outside a set of isolated points, the function \(\phi\) is \({\mathbb{R}}\)-analytic and satisfies \(\phi_{xx}\phi_{yy}-\phi^ 2_{xy}\neq 0.\) Further p-harmonic functions of the form \(r^ kf(\theta)\), where \(x+iy=re^{i\theta},\) are studied in the case where \(p>2\), \(k>(p-2)/(p-1)\); these functions are used to illustrate the representation theorems of Stoilov type for such functions.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 31B30
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 4077831
zbMATH Keywords: p-harmonic
zbMATH Keywords: Sobolev space
zbMATH Keywords: survey
zbMATH Keywords: analytic
zbMATH Keywords: representation
MaRDI profile type: Publication
full work available at URL: 0022-0396(88)90022-8
OpenAlex ID: W1997961641
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description: scientific article; zbMATH DE number 4077831