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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation


Authors: David Kalaj and Miroslav Pavlović
Journal: Trans. Amer. Math. Soc. 363 (2011), 4043-4061
MSC (2010): Primary 30C62
DOI: https://doi.org/10.1090/S0002-9947-2011-05081-6
Published electronically: March 23, 2011
MathSciNet review: 2792979
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Abstract: Let $\mathcal {QC}(K,g)$ be a family of $K$-quasiconformal mappings of the open unit disk onto itself satisfying the PDE $\Delta w =g$, $g\in C(\overline {\mathbb {U}})$, $w(0)=0$. It is proved that $\mathcal {QC}(K,g)$ is a uniformly Lipschitz family. Moreover, if $|g|_\infty$ is small enough, then the family is uniformly bi-Lipschitz. The estimations are asymptotically sharp as $K \to 1$ and $|g|_\infty \to 0$, so $w\in \mathcal {QC}(K,g)$ behaves almost like a rotation for sufficiently small $K$ and $|g|_\infty$.


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Additional Information

David Kalaj
Affiliation: Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put b.b. 81000 Podgorica, Montenegro
MR Author ID: 689421
Email: davidkalaj@gmail.com

Miroslav Pavlović
Affiliation: Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia
Email: pavlovic@matf.bg.ac.rs

Keywords: Quasiconformal harmonic maps, Lipschitz condition
Received by editor(s): May 7, 2008
Received by editor(s) in revised form: April 12, 2009
Published electronically: March 23, 2011
Article copyright: © Copyright 2011 American Mathematical Society