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.