To the problem of determining a mapping from its normalized Jacobi matrix
type: Item
instance of: scholarly article
title: To the problem of determining a mapping from its normalized Jacobi matrix
zbMATH Open document ID: 0836.30015
DOI: 10.1007/BF00971399
author: I. V. Zhuravlev
published in: Siberian Mathematical Journal
publication date: 1996-01-15T00:00:00Z
review text: Let \(D\) be a domain of \(\mathbb{R}^n\) \((n \geq 2)\) and \(f : D \to\mathbb{R}^n\) a mapping, whose coordinate functions \(f^i\) \((i = 1, \dots, n)\) belong to the Sobolev space \(W^1_{n, \text{loc}} (D)\), its Jacobi matrix \(f'\) is defined a.e. in \(D\) and \(J(f,x) = \text{det} f' (x)\). \(f\) is said to be a mapping with bounded distortion if it is continuous and satisfies the system of differential equations \[ f'(x) = K(x) |J(x,f) |^{1/n}, \tag{1} \] a.e in \(D\), where \(\text{det} K(x)\) does not change its sign in \(D\), \(|\text{det} K(x) |= 1\) a.e. in \(D\) and there exists a constant \(C \geq 1\) such that \(|K |_{\infty, D} \leq C\). The author, in a previous paper [Sib. Mat. Zh. 33, 53-61 (1992; Zbl 0806.26008)], established some necessary conditions on the normalized Jacobi matrix \(K(x) = f'(x) \{|J(x,f) |^{1/n}\}\) (considered as a quasiconformal characteristic) if \(f \in C^3\) with \(J(x,f) \neq 0\). In the paper under review, the author proves that these conditions still hold for the more general case of mappings with bounded distortion. He obtains also an existence theorem for the solutions of system (1), allowing to obtain the mapping \(f\) from \(K\). Finally, he shows that the existence of a constant \(K_0\) such that \(|K(x) |\leq K_0\) a.e. in \(D\) implies that \(f\) is locally homeomorphic, hence, on account of a result of \textit{V. A. Zorich} [Mat. Sb., N. Ser. 74(116), 417-433 (1967; Zbl 0181.08701)], if \(D=\mathbb{R}^n,f\) is quasiconformal in \(\mathbb{R}^n\).
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 825557
reviewed by: Petru Caraman
MaRDI profile type: Publication
cites work: Q5563548
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cites work: Some integration problems in almost-complex and complex manifolds
cites work: Harmonic integrals on strongly pseudo-convex manifolds. I
cites work: Q4748438
cites work: Q4748439
cites work: Q3889369
cites work: Q5521609
full work available at URL: https://doi.org/10.1007/bf00971399
OpenAlex ID: W178150576
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description: scientific article; zbMATH DE number 825557