\(n\)-harmonic morphisms in space are Möbius transformations
type: Item
instance of: scholarly article
title: \(n\)-harmonic morphisms in space are Möbius transformations
zbMATH Open document ID: 0801.31004
author: Juan J. Manfredi
author: Vincenzo Vespri
published in: Michigan Mathematical Journal
publication date: 1994-12-04T00:00:00Z
review text: A mapping \(f\) of an open subset of \(\mathbb{R}^ n\) into \(\mathbb{R}^ n\) is called a \(p\)-harmonic morphism if \(u \circ f\) is \(p\)-harmonic in the preimage \(f^{-1}(d)\) whenever \(u\) is \(p\)-harmonic in \(D\). Recall that \(u\) is \(p\)-harmonic in an open set \(\Omega\) if \(u \in C(\Omega) \cap W^{1,p}(\Omega)\) satisfies the \(p\)-Laplace equation \[ \text{div}(| \nabla u|^{p - 2} \nabla u)=0 \] in \(\Omega\); here \(1 < p < \infty\). The case \(p = 2\) is the classical Laplacian; then harmonic morphisms are known to be analytic or antianalytic mappings in the planar case, and similarity transformations (i.e. compositions of rotations and translations) in higher dimensions, and vice versa. For a general \(p\), it is easily seen that the \(p\)-Laplacian is invariant under similarity transformations and, moreover, the \(n\)-Laplacian is invariant under all 1-quasiregular mappings (i.e. Möbius maps if \(n \geq 3\)). Therefore these mappings are \(p\)-harmonic morphisms. In the present paper the authors show that a discrete \(p\)-harmonic morphism necessarily is 1-quasiregular if \(p = n\) and a similarity if \(p \neq n\). The paper complements the paper [\textit{J. Heinonen, T. Kilpelaeinen} and \textit{O. Martio}, Nagoya Math. J. 125, 115-140 (1992; Zbl 0776.31007)], where harmonic morphisms between nonlinear harmonic spaces are characterized.
Mathematics Subject Classification ID: 31C45
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 575626
zbMATH Keywords: quasiregular mappings
zbMATH Keywords: harmonic morphisms
zbMATH Keywords: \(n\)-Laplacian
zbMATH Keywords: \(p\)-harmonic morphisms
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004920
OpenAlex ID: W1987279190
DOI: 10.1307/MMJ/1029004920
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description: scientific article; zbMATH DE number 575626