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

TL;DR

This work proves that a harmonic $K$-quasiconformal map $f:D\to\Omega$ between Lyapunov ($C^{1,\alpha}$) domains with $\alpha\in(0,1)$ is globally Lipschitz on $\overline D$, extending prior results to general $C^{1,\alpha}$ source domains. The authors develop a boundary iteration scheme: starting from a boundary Hölder control dictated by quasiconformality, they improve the Hölder regularity of the normal component via $C^{1,\alpha}$ graph representations of $\partial\Omega$, convert this into near-boundary gradient bounds using a basepoint boundary Hölder-to-gradient estimate, and propagate the control to the full differential through quasiconformality. An exponent-improvement/iteration strategy is then used to achieve bounded $|Df|$ up to the boundary, enabling a global Lipschitz estimate. The approach combines boundary flattening, harmonic-measure bounds, and distortion theory within HQC mappings, and provides a self-contained framework for boundary regularity and co-Lipschitz-type phenomena in higher dimensions. This advances the understanding of Lipschitz behavior for HQC maps and broadens applicability to domains with $C^{1,\alpha}$ boundaries. The results have potential implications for geometric function theory and PDE-driven mapping problems in non-smooth domains.

Abstract

We prove that every sense-preserving harmonic $K$--quasiconformal homeomorphism $f\colon D\toΩ$ between Lyapunov domains (equivalently, bounded $C^{1,α}$ domains) in $\mathbb{R}^n$, $α\in(0,1]$, is globally Lipschitz on $\overline D$. The argument is based on a boundary iteration scheme: an initial Hölder modulus for the boundary trace (coming from quasiconformality) is improved via the $C^{1,α}$ graph representation of $\partialΩ$, yielding higher Hölder regularity for the normal component. This boundary gain is converted into a near-boundary gradient bound for harmonic functions through a basepoint boundary Hölder-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 Hölder continuity, distortion of derivatives, and boundary-to-interior propagation principles that enter the iteration.

Figures (1)

• Figure 1: Local geometry near a $C^{1,\alpha}$ boundary patch. The boundary of $\Omega_{x_0}=D\cap B(x_0,r_0)$ is decomposed as $\partial\Omega_{x_0}=\Gamma\cup\Sigma$, where $\Gamma=\partial D\cap B(x_0,r_0)$ and $\Sigma=\partial B(x_0,r_0)\cap \overline{D}.$

Theorems & Definitions (7)

• Theorem 2.1: Lipschitz regularity for harmonic quasiconformal maps
• Lemma 3.1: Basepoint boundary Hölder at $x_0$ $\Rightarrow$ interior Hölder and gradient control along the normal
• proof
• Lemma 3.2: Addendum to Lemma \ref{['lem:anchored_holder_to_grad_normal']}: the case $\mu>1$
• proof : Proof
• proof : Proof of Theorem \ref{['thm:lipschitz_C1a']}
• Remark 4.1