Harmonic mappings and distance function

TL;DR

This paper proves that quasiconformal harmonic mappings between certain smooth plane domains are bi-Lipschitz, extending previous results by utilizing the boundary distance function and relaxing boundary regularity conditions.

Contribution

The paper extends earlier work by showing bi-Lipschitz properties of harmonic quasiconformal maps with less restrictive boundary smoothness, using the boundary distance function.

Findings

Quasiconformal harmonic mappings are bi-Lipschitz under specified boundary conditions.

Extension of previous results to domains with $C^{1,eta}$ boundaries.

Use of boundary distance function is key to the proof.

Abstract

We prove the following theorem: every quasiconformal harmonic mapping between two plane domains with $C^{1,\alpha}$ ($\alpha<1$), respectively $C^{1,1}$ compact boundary is bi-Lipschitz. The distance function with respect to the boundary of the image domain is used. This in turn extends a similar result of the author in \cite{kalajan} for Jordan domains, where stronger boundary conditions for the image domain were needed.