Quasiconformal harmonic mappings between Dini's smooth Jordan domains

TL;DR

This paper proves that harmonic quasiconformal mappings between Jordan domains with Dini's smooth boundaries are Lipschitz continuous, extending previous results under weaker boundary regularity assumptions.

Contribution

It establishes the Lipschitz continuity of harmonic quasiconformal maps between Dini smooth Jordan domains, generalizing prior results with stronger boundary conditions.

Findings

Harmonic quasiconformal maps are Lipschitz under Dini boundary conditions.

Extends previous results to weaker boundary regularity.

Results are optimal, matching conditions for conformal maps.

Abstract

Let $D$ and $\Omega$ be Jordan domains with Dini's smooth boundaries and and let $f:D\mapsto \Omega$ be a harmonic homeomorphism. The object of the paper is to prove the following result: If $f$ is quasiconformal, then $f$ is Lipschitz. This extends some recent results, where stronger assumptions on the boundary are imposed, and somehow is optimal, since it coincides with the best condition for Lipschitz behavior of conformal mappings in the plane and conformal parametrization of minimal surfaces.