Quasiconformal harmonic mappings between Dini-smooth Jordan domains

David Kalaj

Vol. 276 (2015), No. 1, 213–228

DOI: 10.2140/pjm.2015.276.213

Abstract

Let $D$ and $\Omega$ be Jordan domains with Dini-smooth boundaries. We prove that if $f:D\to \Omega$ is a harmonic homeomorphism and $f$ is quasiconformal, then $f$ is Lipschitz. This extends some recent results, where stronger assumptions on the boundary are imposed. Our result is optimal in that it coincides with the best condition for Lipschitz behavior of conformal mappings in the plane and conformal parametrizations of minimal surfaces.

Keywords

planar harmonic mappings, quasiconformal, Dini-smooth

Mathematical Subject Classification 2010

Primary: 30C55

Secondary: 31C05

Milestones

Received: 7 February 2014

Revised: 17 December 2014

Accepted: 19 December 2014

Published: 1 July 2015