On p-harmonic functions in the complex plane and curvature
type: Item
instance of: scholarly article
title: On p-harmonic functions in the complex plane and curvature
zbMATH Open document ID: 0667.31008
DOI: 10.1007/BF02778033
author: Peter Lindqvist
published in: Israel Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let u be a p-harmonic function, i.e. a weak solution of a p-harmonic equation \(div(| \nabla u|^{p-2}\nabla u)=0\), \(1<p<\infty\), in a plane domain. The author investigated some interesting p-harmonic analogues of the properties of harmonic functions \((p=2)\). For example let u be a p-harmonic function in \(G\subset {\mathbb{R}}^ 2\) and \(x_ 3=u(x_ 1,x_ 2)\) represents a p-harmonic surface in \({\mathbb{R}}^ 3.\) Theorem 5.3: The Gauss curvature \(K\leq 0\) for a p-harmonic surface. Either \(K=0\) at most at isolated points, or the surface is a plane. Theorem 5.5: If u is p-harmonic in the whole plane, then the total curvature of the corresponding p-harmonic surface is \(\iint K d\sigma \leq -2\pi,\) except when the surface is a plane.
Mathematics Subject Classification ID: 31C05
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 53A05
zbMATH DE Number: 4091843
zbMATH Keywords: p-harmonic function
zbMATH Keywords: harmonic functions
zbMATH Keywords: p-harmonic surface
zbMATH Keywords: Gauss curvature
zbMATH Keywords: total curvature
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4091843