Harmonic quasiconformal mappings between ${\mathcal{C}}^1$ smooth Jordan domains

Abstract

We prove the following result. If $f$ is a harmonic quasiconformal mapping between two Jordan domains $D$ and $\Omega$ having ${\mathcal{C}}^1$ boundaries, then the function $f$ is globally Hölder continuous for every $\alpha <1$ but it is not necessarily Lipschitz in general. This result extends and improves a classical theorem of S. Warschawski for conformal mappings.

A correction to this paper is available.