Quasiconformal maps with controlled Laplacian

We establish that every $K$-quasiconformal mapping of $w$ of the unit disk $\ID$ onto a $C^2$-Jordan domain $Ω$ is Lipschitz provided that $Δw\in L^p(\ID)$ for some $p>2$. We also prove that if in this situation $K\to 1$ with $\|Δw\|_{L^p(\ID)}\to 0$, and $Ω\to \ID$ in $C^{1,α}$-sense with $α>1/2,$ 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.

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