On mappings of plane domains by solutions of second-order elliptic equations
type: Item
instance of: scholarly article
title: On mappings of plane domains by solutions of second-order elliptic equations
zbMATH Open document ID: 1405.30048
DOI: 10.3103/S1066369X18080042
author: A. B. Zaitsev
published in: Russian Mathematics
publication date: 2018-10-12T00:00:00Z
review text: In the interesting paper under review, the author obtains sufficient conditions for one-to-one solvability of the second-order partial differential equation \[ \overline{\partial}\partial_\beta u=0 \] in a plane Jordan domain. Here \(\overline{\partial}\) is the Cauchy-Riemann operator, while \(\partial_\beta=\frac{\partial}{\partial x}+i\beta\frac{\partial}{\partial y}\) with \(\beta\in (-1,0).\) The main result, regarding a continuous one-to-one and orientation-keeping map of the boundary of a Jordan domain to the rectifiable boundary of some other Jordan domain, asserts the following: If the Cauchy integral with a measure generated by this map is bounded by some constant in the exterior domain, then the solution to the corresponding Dirichlet problem in the domain with this boundary function maps these domains in an one-to-one way. The proof relies on integral representations of the solutions and on Fredholm properties of the integral equations on the boundary.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 30G20
Mathematics Subject Classification ID: 31A20
zbMATH DE Number: 6952509
zbMATH Keywords: elliptic operator
zbMATH Keywords: Jordan domain
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: one-to-one map
MaRDI profile type: Publication
cites work: Nonsymmetric systems on nonsmooth planar domains
cites work: Q4828548
cites work: Elliptic systems and material interpenetration
cites work: Mappings by the solutions of second-order elliptic equations
cites work: Q5527180
full work available at URL: s1066369x18080042
OpenAlex ID: W2883433912
Wikidata QID: Q129488209
description: scientific article; zbMATH DE number 6952509