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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
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 $ \vert g\vert _\infty$ is small enough, then the family is uniformly bi-Lipschitz. The estimations are asymptotically sharp as $ K \to 1$ and $ \vert g\vert _\infty\to 0$, so $ w\in \mathcal{QC}(K,g)$ behaves almost like a rotation for sufficiently small $ K$ and $ \vert g\vert _\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
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

DOI: https://doi.org/10.1090/S0002-9947-2011-05081-6
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