On the Riemann-Hilbert problem for the Beltrami equations in quasidisks
type: Item
instance of: scholarly article
title: On the Riemann-Hilbert problem for the Beltrami equations in quasidisks
zbMATH Open document ID: 1338.30028
author: Artem S. Efimushkin
author: Vladimir I. Ryazanov
published in: Journal of Mathematical Sciences (New York)
publication date: 2016-01-19T00:00:00Z
review text: Let \(D\) be a domain in the complex plane \(\mathbb C\) and let \(\mu:D\to\mathbb C\) be a measurable function with \(|\mu(z)|<1\) a.e. The Beltrami equation in \(D\) with coefficient \(\mu\) is the equation \[ f_{\bar{z}} =\mu(z) f_z, \tag{1} \] where \(f_{\bar{z}} =\bar{d}f=(f_x+if_y )/2\), \( f_z=df=(f_x-if_y )/2\), \(z=x+iy\), \(f_x\) and \(f_y\) are the partial derivatives of the function \(f:D\to\mathbb C \) with respect to \(x\) and \(y\), respectively. Equation (1) is called non-degenerate if \(\|\mu\|_\infty<1\). The homeomorphic solutions of the non-degenerate Beltrami equations (1) with generalized derivatives are called quasi-conformal mappings. The images of the unit circle \(D=\{z\in C; |z|<1\}\) under quasi-conformal mappings of \(\mathbb C\) onto itself are called quasidisks, and their boundaries are called quasicircles or quasi-conformal curves. In the paper under review, for the nondegenerate Beltrami equation in quasidisks, the existence of regular solutions of the Riemann-Hilbert problem with coefficients of bounded variation and boundary data that are measurable with respect to the absolute harmonic measure (logarithmic capacity) is proved.
Mathematics Subject Classification ID: 30E25
Mathematics Subject Classification ID: 30C62
zbMATH DE Number: 6532884
zbMATH Keywords: Riemann-Hilbert problem
zbMATH Keywords: Beltrami equation
zbMATH Keywords: quasidisks
zbMATH Keywords: quasiconformal curves
zbMATH Keywords: logarithmic capacity
reviewed by: Nino Manjavidze
MaRDI profile type: Publication
full work available at URL: s10958-015-2621-0
OpenAlex ID: W2286760961
cites work: Angles and quasiconformal mappings in space
cites work: Angles and Quasiconformal Mappings†
cites work: Q5512337
cites work: CURVILINEAR CLUSTER SETS OF ARBITRARY FUNCTIONS
cites work: Q5338374
cites work: Q5583306
cites work: Geometric measure theory.
cites work: Q4003731
cites work: Q3236000
cites work: Q4433734
cites work: Q5574105
cites work: The Beltrami equation. A geometric approach
cites work: Q4856605
cites work: On the boundary behavior of solutions of the Beltrami equations
cites work: On the Dirichlet problem for the Beltrami equations in finitely connected domains
cites work: Q5685358
cites work: Moduli in Modern Mapping Theory
cites work: Q3844606
cites work: Q3265777
cites work: Q4405418
cites work: Q3264943
cites work: Q4115333
cites work: On the Riemann-Hilbert problem IV
cites work: Q4932610
cites work: Q5529726
cites work: Q5292491
cites work: Q3281267
cites work: Q3254471
DOI: 10.1007/S10958-015-2621-0
description: scientific article; zbMATH DE number 6532884