Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 811-830

QUASICONFORMAL AND LIPSCHITZ HARMONIC MAPPINGS OF THE UNIT DISK ONTO BOUNDED CONVEX DOMAINS

Dariusz Partyka and Ken-ichi Sakan

The John Paul II Catholic University of Lublin, Institute of Mathematics and Computer Science
Al. Raclawickie 14, P.O. Box 129, 20-950 Lublin, Poland; partyka 'at' kul.lublin.pl, and
The State School of Higher Education in Chelm, Institute of Mathematics and Information Technology
Pocztowa 54, 22-100 Chelm, Poland

Osaka City University, Graduate School of Science, Department of Mathematics
Sugimoto, Sumiyoshi-ku, Osaka, 558, Japan; ksakan 'at' sci.osaka-cu.ac.jp

Abstract. We study the Lipschitz property of a harmonic injective and sense-preserving mapping F of the unit disk D onto a bounded convex domain Ω in the complex plane C. In particular we show that F is bi-Lipschitz iff F is quasiconformal and Lipschitz. To this end we establish some auxiliary properties of harmonic mappings dealing with the boundary radial limiting values of the formal derivatives ∂F and \bar ∂F.

2010 Mathematics Subject Classification: Primary 30C55, 30C62.

Key words: Harmonic mappings, Lipschitz mappings, Poisson integral, Jacobian, quasiconformal mappings.

Reference to this article: D. Partyka and K. Sakan: Quasiconformal and Lipschitz harmonic mappings of the unit disk onto bounded convex domains. Ann. Acad. Sci. Fenn. Math. 39 (2014), 811-830.

Full document as PDF file

doi:10.5186/aasfm.2014.3940