Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Published electronically: March 23, 2011
MathSciNet review: 2792979
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 30C62

Retrieve articles in all journals with MSC (2010): 30C62

Additional Information

David Kalaj
Affiliation: Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put b.b. 81000 Podgorica, Montenegro

Miroslav Pavlović
Affiliation: Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia

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