Quasiconformal harmonic mappings between two doubly connected domains in the plane

Abstract

It is known for some time that there exists an energy-minimal diffeomorphism between two doubly-connected domains $\Omega$ and $D$ provided that $\mathrm{Mod}(\Omega)\le \mathrm{Mod}{D}$ and that diffeomorphism is harmonic \cite{tedi}. In this note we give a short proof of the fact that for given annuli $\Omega$ and $D$ satisfying the condition $\mathrm{Mod}(\Omega)\le \mathrm{Mod}(D)$ there exist a $K-$quasiconformal harmonic diffeomorphism $f:\Omega\onto D$, where $K=K(\tau)$, $\tau=\mathrm{Mod}(\Omega)/ \mathrm{Mod}(D)$ and $\lim_{\tau\to 1}K(\tau)=1$.