Lipschitz spaces and harmonic mappings

TL;DR

This paper extends the bi-Lipschitz property of quasiconformal harmonic mappings to non-convex Jordan domains with smooth boundaries, removing previous convexity restrictions.

Contribution

It proves that quasiconformal harmonic mappings between Jordan domains with smooth boundaries are bi-Lipschitz without requiring convexity.

Findings

Bi-Lipschitz property holds for non-convex domains

Extends previous results to more general boundary conditions

Maintains bi-Lipschitz continuity under relaxed assumptions

Abstract

In \cite{kamz} the author proved that every quasiconformal harmonic mapping between two Jordan domains with $C^{1,\alpha}$, $0<\alpha\le 1$, boundary is bi-Lipschitz, providing that the domain is convex. In this paper we avoid the restriction of convexity. More precisely we prove: any quasiconformal harmonic mapping between two Jordan domains $\Omega_j$, $j=1,2$, with $C^{j,\alpha}$, $j=1,2$ boundary is bi-Lipschitz.