Lipschitz regularity of harmonic quasiconformal maps between Lyapunov domains in $\mathbb{R}^n$

TL;DR

This paper proves that harmonic quasiconformal maps between Lyapunov domains in or all ormally, bounded unctions are globally Lipschitz, using boundary iteration and harmonic measure estimates.

Contribution

It establishes the Lipschitz regularity of harmonic quasiconformal maps between Lyapunov domains, extending boundary regularity results through an iterative boundary-to-interior approach.

Findings

Harmonic quasiconformal maps are globally Lipschitz on domain closures.

Boundary regularity improves via an iterative Hlder-to-gradient scheme.

The method combines boundary flattening, harmonic measure bounds, and quasiconformal distortion control.

Abstract

We prove that every sense-preserving harmonic $K$--quasiconformal homeomorphism $f\colon D\to\Omega$ between Lyapunov domains (equivalently, bounded $C^{1,\alpha}$ domains) in $\mathbb{R}^n$, $\alpha\in(0,1]$, is globally Lipschitz on $\overline D$. The argument is based on a boundary iteration scheme: an initial H\"older modulus for the boundary trace (coming from quasiconformality) is improved via the $C^{1,\alpha}$ graph representation of $\partial\Omega$, yielding higher H\"older regularity for the normal component. This boundary gain is converted into a near-boundary gradient bound for harmonic functions through a basepoint boundary H\"older-to-gradient estimate obtained by flattening the boundary and using local harmonic-measure bounds. Quasiconformality then propagates the resulting control from one component to the full differential, and iteration gives boundedness of $|Df|$ up to the boundary. Along the way we briefly survey several standard tools from the theory of quasiconformal harmonic mappings (QCH), including boundary H\"older continuity, distortion of derivatives, and boundary-to-interior propagation principles that enter the iteration.