The Inverse of a σ(z)-Harmonic Diffeomorphism

A new kind of functional, analogous to the Douglas–Dirichlet functional, is defined as E ′ [ f ] = ∫ ∫ Ω σ ( z ) ( | f z | 2 + | f z ¯ | 2 ) d x d y for f ∈ C 2 on Ω with a conformal metric density σ ( z ) . A critical point of this new functional is said to be a σ ( z ) -harmonic mapping. We consid...

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Bibliographic Details
Published in:Computational methods and function theory 2017-12, Vol.17 (4), p.653-662
Main Authors: Chunying, Hu, Qingtian, Shi
Format: Article
Language:English
Subjects:
Analysis
Computational Mathematics and Numerical Analysis
Critical point
Dirichlet problem
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Online Access:Get full text
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Summary:A new kind of functional, analogous to the Douglas–Dirichlet functional, is defined as E ′ [ f ] = ∫ ∫ Ω σ ( z ) ( | f z | 2 + | f z ¯ | 2 ) d x d y for f ∈ C 2 on Ω with a conformal metric density σ ( z ) . A critical point of this new functional is said to be a σ ( z ) -harmonic mapping. We consider the harmonicity of the inverse function of a σ ( z ) -harmonic diffeomorphism and obtain a necessary and sufficient condition, which improves on the corresponding result for Euclidean harmonic mappings. In addition, a property of the inverse function of ρ -harmonic mappings is investigated and an example is given.
ISSN:1617-9447
2195-3724
DOI:10.1007/s40315-017-0202-6