Representation of a p-harmonic function near a critical point in the plane
type: Item
instance of: scholarly article
title: Representation of a p-harmonic function near a critical point in the plane
zbMATH Open document ID: 0702.35049
DOI: 10.1007/BF02568483
author: Gunnar Aronsson
published in: Manuscripta Mathematica
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155460
review text: Let \(\Omega\) be a domain in \({\mathbb{R}}^ 2\) and let \(1<p<\infty\). A p- harmonic function \(\phi\) on \(\Omega\) is a weak solution of the equation \(div(| \nabla \phi |^{p-2}\nabla \phi)=0.\) Given such a function \(\phi\), there is an \(\alpha >0\) such that \(\phi\) belongs to \(C^{1,\alpha}\) on any compact subset of \(\Omega\). In fact, \(\phi\) is real analytic on \(\Omega\setminus E\), where E is the set of critical points of \(\phi\). It is the nature of \(\phi\) near critical points (which are known to be isolated) that is investigated in this paper. Using ``stream functions'' and the ``hodograph transformation'', the author establishes a representation theorem for \(\phi\) on a neighbourhood of a critical point \(z_ 0\). Among the consequences of this result is the determination of the exact Hölder exponent of \(\nabla \phi\) near \(z_ 0\) as a function of p and the ``order'' of the critical point.
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 31C45
Mathematics Subject Classification ID: 35J70
zbMATH DE Number: 4150788
zbMATH Keywords: p-harmonic function
zbMATH Keywords: analytic
zbMATH Keywords: critical points
zbMATH Keywords: stream functions
zbMATH Keywords: hodograph transformation
zbMATH Keywords: representation theorem
zbMATH Keywords: exact Hölder exponent
reviewed by: Stephen J. Gardiner
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4150788