Quasiregular mappings of several \(n\)-dimensional variables
type: Item
instance of: scholarly article
title: Quasiregular mappings of several \(n\)-dimensional variables
zbMATH Open document ID: 0809.30014
DOI: 10.1007/BF00975169
published in: Siberian Mathematical Journal
publication date: 1994-11-08T00:00:00Z
review text: Let \(U\) be a domain in \((\mathbb{R}^ n)^ k\) and let \(f=(f^ 1, f^ 2, \dots, f^ m)\) be a mapping from \(U\) to \((\mathbb{R}^ n)^ m\) with Jacobi matrix \(f'(x)\) defined almost everywhere in \(U\). A mapping \(f\) is said to be \(K\)-quasiregular mapping of several \(n\)-dimensional variables if the following inequality holds almost everywhere in \(U\): \[ \sum_{i,j} | f_{ij}'(x) |^ n \leq K \cdot n^{n \over 2} \sum_{i,j} J_{ij} (x,f), \] where \(J_{ij} (x,f) = \text{det} f_{ij}'(x)\) and \(| A | = (\sum_{i,j} a^ 2_{ij})^{1/2}\) for any matrix \(A \in \mathbb{R}^{s \times t}\), \(A = (a_{ij})_{i=1,2, \dots, s;j = 1,2, \dots,t}\). The family of such mappings are denoted by \(Q(K)\). In this paper some properties of such mappings are investigated. In particularly it is proved that for \(K \geq 1\) the family \(Q(K)\) has the following properties. \(1^ \circ\) If \(f_ s:U \to (\mathbb{R}^ n)^ m\), \(s = 1,2, \dots\), is a sequence of mappings in \(Q(K)\) converging to a mapping \(f_ 0\) then \(f_ 0 \in Q(K)\). \(2^ \circ\) Let \(f=(f_ 1, f_ 2, \dots, f_ n) : U \to \mathbb{R}^ n\) be a mapping of the class \(Q(K)\), \((m=1)\), then there are some polynomials of a special type \(Q_ j\), \(P_ j\), \(j=1,2, \dots,n\) such that \[ f_ j(x) = P_ j(x)/Q_ j(x), \quad x \in U, \quad j = 1,2, \dots,n; \] moreover, each \(Q_ j\) nonvanishes in \(U\).
reviewed by: Jan Stankiewicz
Mathematics Subject Classification ID: 30C65
zbMATH DE Number: 652176
zbMATH Keywords: quasiregular mapping
author: N. S. Dairbekov
MaRDI profile type: Publication
cites work: Q3947877
cites work: Q3994667
cites work: Analytical foundations of the theory of quasiconformal mappings in R^n
cites work: Quasiconformal 4-manifolds
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description: scientific article; zbMATH DE number 652176