On the theory of convergence and compactness for Beltrami equations with constraints of set-theoretic type
type: Item
instance of: scholarly article
title: On the theory of convergence and compactness for Beltrami equations with constraints of set-theoretic type
zbMATH Open document ID: 1267.30061
published in: Ukrainian Mathematical Journal
publication date: 2012-11-23T00:00:00Z
review text: Let \(D\) be a domain in the complex plane \(\mathbb C\), let \(\mu (z) : D \rightarrow \mathbb C\) be a measurable function such that \(|\mu (z)| < 1\) almost everywhere, \(z = x+iy\), \(f\overline{z} =\frac{ f_x +if_y}{2}\), \(f_z =\frac{f_x - if_y}{2}\) and \(K_{\mu} (z) = (1 + |\mu (z)|)/(1 - |\mu (z)|)\). Then the equation \(f\overline{z} = \mu (z)f_z\) is called a Beltrami equation. In particular, if \(\text{esssup}K_{\mu} (z) = \infty\), then it is called degenerate. In the paper under review, the author establishes several theorems on the convergence and compactness for the regular solutions of the degenerate Beltrami equations with constraints of the set-theoretic type imposed on the complex coefficient \(\mu (z)\). As applications, variations for the classes of regular solutions of degenerate Beltrami equations are constructed.
reviewed by: Yu-Ming Chu
Mathematics Subject Classification ID: 30C62
zbMATH DE Number: 6110037
zbMATH Keywords: convergence
zbMATH Keywords: Beltrami equation
zbMATH Keywords: regular solution
zbMATH Keywords: complex coefficient
zbMATH Keywords: compactness
zbMATH Keywords: dilatation
author: T. V. Lomako
MaRDI profile type: Publication
full work available at URL: s11253-012-0587-3
OpenAlex ID: W2064516947
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DOI: 10.1007/S11253-012-0587-3
description: scientific article; zbMATH DE number 6110037