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On the existence of harmonic mappings between doubly connected domains

Part of: Two-dimensional theory Variational problems in infinite-dimensional spaces Geometric function theory

Published online by Cambridge University Press:  22 April 2018

Leonid V. Kovalev  and
Liulan Li
Leonid V. Kovalev
Affiliation:
Department of Mathematics, Syracuse University, 215 Carnegie, Syracuse, NY 13244, USA ([email protected])
Liulan Li
Affiliation:
Hengyang Normal University, Hengyang, Hunan 421002, People's Republic of China ([email protected])
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Abstract

While the existence of conformal mappings between doubly connected domains is characterized by their conformal moduli, no such characterization is available for harmonic diffeomorphisms. Intuitively, one expects their existence if the domain is not too thick compared to the codomain. We make this intuition precise by showing that for a Dini-smooth doubly connected domain Ω* there exists a ε > 0 such that for every doubly connected domain Ω with ModΩ* < ModΩ < ModΩ* + ε there exists a harmonic diffeomorphism from Ω onto Ω*.

Keywords

harmonic mappingconformal modulusaffine modulusdoubly connected domain

MSC classification

Secondary: 31A05: Harmonic, subharmonic, superharmonic functions 58E20: Harmonic maps , etc. 30C20: Conformal mappings of special domains
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 3 , June 2018 , pp. 619 - 628
Copyright
Copyright © Royal Society of Edinburgh 2018 

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