Quasiconformal maps with controlled Laplacian

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.