On the Dirichlet problem for the Beltrami equations in finitely connected domains
type: Item
instance of: scholarly article
title: On the Dirichlet problem for the Beltrami equations in finitely connected domains
zbMATH Open document ID: 1276.30039
author: D. A. Kovtonyuk
author: I. V. Petkov
author: Vladimir I. Ryazanov
published in: Ukrainian Mathematical Journal
publication date: 2013-02-01T00:00:00Z
review text: The Beltrami equation has the form \(f_{\bar z}=\mu(z) f_z\), where \(\mu\) is a measurable complex-valued function such that \(|\mu|<1\) almost everywhere in some domain \(D\) in the complex plane. A map \(f: D\to\mathbb C\) is called a regular solution of the Beltrami equation if it is continuous, discrete and open, is locally in the Sobolev class \(W^{1,1}\), satisfies the equation a.e., and has Jacobian determinant that is nonzero a.e. The authors give several sufficient conditions for the existence of regular solutions with prescribed boundary values in a Jordan domain \(D\subset \mathbb C\). To state one of them, let \(\|K_\mu\|_1(a,r)\) denote the average of the dilatation \(K_\mu=(1+|\mu|)/(1-|\mu|)\) over the circle of radius \(r\) centered at \(a\). Suppose that \(K_\mu\) is integrable in \(D\) and the improper integral \(\int_0 \frac{dr}{\|K_\mu(a,r)\|_1(r)}\) diverges for all \(a\in \overline{D}\). Theorem 2 of the present article asserts that under these conditions, for any continuous function \(\varphi : \partial D\to\mathbb R\), the Beltrami equation has a regular solution \(f\) such that \(\lim_{z\to \zeta} \mathrm{Re}\, f(z)=\varphi(z)\) for all \(\zeta\in \partial D\).
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30C75
zbMATH DE Number: 6133800
zbMATH Keywords: Beltrami equation
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Jordan domain
zbMATH Keywords: dilatation
reviewed by: Leonid V. Kovalev
MaRDI profile type: Publication
full work available at URL: s11253-012-0699-9
OpenAlex ID: W1990790566
cites work: Q3268429
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DOI: 10.1007/S11253-012-0699-9
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description: scientific article; zbMATH DE number 6133800