Lipschitz spaces and harmonic mappings

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.