Stability of classes of quasiregular mappings in several spatial variables
type: Item
instance of: scholarly article
title: Stability of classes of quasiregular mappings in several spatial variables
zbMATH Open document ID: 0856.30019
DOI: 10.1007/BF02113918
published in: Siberian Mathematical Journal
publication date: 1997-02-17T00:00:00Z
review text: Let \((\mathbb{R}^n)^k = \mathbb{R}^n \times \mathbb{R}^n \times \cdots \times \mathbb{R}^n\), \((k\)-times), \(x \in (\mathbb{R}^n)^k\) may be written as \(x = (x_1, x_2, \dots, x_k)\) where \(x_j \in \mathbb{R}^n \), \(j = 1,2, \dots, k\). For a given domain \(U \subset (\mathbb{R}^n)^k\) we introduce a function \(f = (f_1, f_2, \dots, f_m) : U \to (\mathbb{R}^n)^m\) of the class \(W^1_{n, \text{loc}} (U, (\mathbb{R}^n)^m)\) with Jacobi matrix \[ f'(x) = \bigl[ f_{ij}'(x) \bigr]_{i = 1, 2, \dots, m; j = 1, 2, \dots, k} \quad \text{where} \quad \bigl[ f_{ij}' \bigr] = \left[ {\partial f^i \over \partial x_j} \right] \in \mathbb{R}^{n \times n} \] are the matrices of a dimension \(n \times n\). Definition A mapping \(f : U \to (\mathbb{R}^n)^m\) is called the quasiregular mapping of several spatial variables if \(1^\circ\) \(f \in W^1_{n, \text{loc}} (U, (\mathbb{R}^n)^m)\); \(2^\circ\) there exists a constant \(K < \infty\) such that \[ \sum_{i,j} |f_{ij}' |^n \leq K n^{n/2} \cdot \sum_{i,j} \text{det} f_{ij}', \quad \text{where} \quad |A |= \bigl |[a_{rs}] \bigr |= \Bigl( \sum_{r,s} a^2_{rs} \Bigr)^{1/2} \] and let \(\text{det} A\) denote the determinant of \(A\). The constant \(K = K(f)\) is called a coefficient of quasiregularity of a mapping \(f\) and the set of all quasiregular mappings such that \(K(f) \leq K\) is denoted by \(G(K) = G^m_{n,k} (K)\). In this paper some properties of such classes of quasiregular mappings of several spatial variables are investigated. This results in the special cases coincide with the earliest results of the author and the results of Kopylov and Reshetnyak.
reviewed by: Jan Stankiewicz
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 26E10
Mathematics Subject Classification ID: 26B10
zbMATH DE Number: 923477
zbMATH Keywords: quasiregular mapping
author: N. S. Dairbekov
MaRDI profile type: Publication
cites work: Quasiregular mappings of several \(n\)-dimensional variables
cites work: Q3994667
cites work: Q4748438
cites work: Q3947877
cites work: Q3673610
full work available at URL: https://doi.org/10.1007/bf02113918
OpenAlex ID: W2086304410
description: scientific article; zbMATH DE number 923477