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

Technical Summary: Lipschitz Regularity of Harmonic Quasiconformal Maps Between Lyapunov Domains

Problem Statement
The paper addresses the regularity of harmonic quasiconformal (HQC) mappings between bounded domains in $\mathbb{R}^n$. Specifically, it investigates whether a sense-preserving harmonic $K$-quasiconformal homeomorphism $f: D \to \Omega$ is globally Lipschitz continuous when the source domain $D$ and target domain $\Omega$ are Lyapunov domains (equivalently, bounded $C^{1,\alpha}$ domains with $\alpha \in (0, 1]$).

While previous work (specifically [11]) established Lipschitz continuity when the source domain was the unit ball $B^n$ and the target had a $C^{1,\alpha}$ boundary, the general case where the source domain $D$ is an arbitrary $C^{1,\alpha}$ domain remained open. The authors aim to extend this result to arbitrary $C^{1,\alpha}$ source and target domains.

Methodology
The proof relies on a boundary iteration scheme that converts initial Hölder continuity of the boundary trace into higher regularity, eventually yielding a bounded gradient up to the boundary. The methodology proceeds through the following stages:

1. Global Hölder Continuity: The authors first establish that $f$ is globally $\beta$-Hölder continuous for some $\beta \in (0, 1)$. This is derived from the fact that bounded $C^{1,\alpha}$ domains are uniform domains (and thus John domains), allowing the application of existing theorems regarding mappings between uniform and John domains.
2. Boundary-to-Interior Gradient Estimates: The core analytic tool is a localized estimate near a boundary point $x_0 \in \partial D$.
   • Lemma 3.1: If the boundary trace of a harmonic function $u$ satisfies a $\mu$-Hölder condition ($0 < \mu < 1$) at $x_0$, then the gradient $|\nabla u(x)|$ along the inward normal ray decays as $t^{\mu-1}$, where $t$ is the distance to the boundary.
   • Lemma 3.2: If the Hölder exponent $\mu > 1$, the gradient estimate becomes uniform (bounded) near the boundary.
   • These lemmas utilize harmonic measure estimates on $C^{1,\alpha}$ domains, flattening the boundary to a graph, and applying standard boundary Lipschitz estimates.
3. Exponent Improvement via Quasiconformality:
   • The authors rotate the coordinate system at a boundary point $q = f(\xi)$ so that the target boundary $\partial \Omega$ is locally represented as a $C^{1,\alpha}$ graph $y_n = \Phi(y')$.
   • The normal component of the map, $f_n$, is constrained by the boundary geometry: $f_n(\eta) = \Phi(\tilde{f}(\eta))$. Since $\Phi$ is $C^{1,\alpha}$ and the tangential components are $\beta$-Hölder, the normal component gains regularity, satisfying a Hölder condition with exponent $\beta_1 = (1+\alpha)\beta$.
   • Quasiconformality is then used to propagate this control from the normal component to the full differential $|Df|$. Specifically, the smallest singular value of the differential controls the full norm, and the boundary gradient estimate for the normal component provides the necessary bound.
4. Iteration: The improved Hölder exponent $\beta_1$ is fed back into the gradient estimate. By iterating this process ($\beta_{m+1} = (1+\alpha)\beta_m$), the exponent eventually exceeds 1. Once $\beta_m > 1$, Lemma 3.2 applies, yielding a uniform bound on $|Df|$ in a collar neighborhood of the boundary.
5. Globalization: The local bounds near the boundary are combined with standard interior gradient estimates (valid since $f$ is harmonic and smooth in the interior) to establish a global bound on $|Df|$.

Key Contributions and Results

• Main Theorem (Theorem 2.1): The authors prove that if $D$ and $\Omega$ are bounded $C^{1,\alpha}$ domains in $\mathbb{R}^n$ ($\alpha \in (0, 1)$), then any harmonic $K$-quasiconformal homeomorphism $f: D \to \Omega$ is globally Lipschitz continuous.
• Extension of Prior Work: This result generalizes the findings of [11], removing the restriction that the source domain must be the unit ball.
• Technical Framework: The paper provides a detailed "boundary iteration scheme" that explicitly links the $C^{1,\alpha}$ regularity of the boundary to the gain in Hölder exponents of the normal component, and subsequently to the boundedness of the full gradient.
• Auxiliary Estimates: The paper establishes precise quantitative estimates (Lemmas 3.1 and 3.2) relating boundary Hölder control to interior gradient bounds for harmonic functions in $C^{1,\alpha}$ domains, distinguishing between the cases $\mu < 1$ and $\mu > 1$.

Significance and Claims
The paper claims to resolve the regularity question for HQC maps between arbitrary Lyapunov domains. The significance lies in:

• Removing Geometric Constraints: By allowing the source domain to be an arbitrary $C^{1,\alpha}$ domain rather than just the unit ball, the result applies to a much broader class of geometric configurations.
• Unified Approach: The argument synthesizes tools from harmonic analysis (harmonic measure, boundary estimates) and quasiconformal geometry (distortion of derivatives, singular values) to propagate regularity from the boundary to the interior.
• Modesty in Scope: The authors frame the work as an improvement upon [11] under the sole assumption of $C^{1,\alpha}$ boundaries for the source. They do not claim to solve the problem for less regular domains (e.g., Lipschitz domains without $C^{1,\alpha}$ smoothness) or to provide bi-Lipschitz constants, focusing strictly on the global Lipschitz property.

The paper concludes by noting that in the planar case, similar results were previously obtained via reduction to the unit disk and conformal mapping theory, but the higher-dimensional proof presented here relies on the intrinsic iteration scheme described above.