Quasiconformal maps with controlled Laplacian

TL;DR

This paper proves that certain quasiconformal maps with controlled Laplacian are Lipschitz, and explores how their Lipschitz constants behave as the maps approach conformality and the domains approach the unit disk.

Contribution

The paper establishes Lipschitz regularity for quasiconformal maps with Laplacian in L^p, and analyzes the asymptotic behavior of their Lipschitz constants as maps become conformal.

Findings

Maps are Lipschitz if Laplacian is in L^p for p>2.

Lipschitz constant approaches 1 as maps become conformal and domains approach the disk.

Provides a quasiconformal analogue of Smirnov's boundary absolute continuity.

Abstract

We establish that every $K$-quasiconformal mapping of $w$ of the unit disk $\ID$ onto a $C^2$-Jordan domain $\Omega$ is Lipschitz provided that $\Delta w\in L^p(\ID)$ for some $p>2$. We also prove that if in this situation $K\to 1$ with $\|\Delta w\|_{L^p(\ID)}\to 0$, and $\Omega \to \ID$ in $C^{1,\alpha}$-sense with $\alpha>1/2,$ then the bound for the Lipschitz constant tends to $1$. In addition, we provide a quasiconformal analogue of the Smirnov absolute continuity result over the boundary.