On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation

Abstract:

Let $\mathcal {QC}(K,g)$ be a family of $K$-quasiconformal mappings of the open unit disk onto itself satisfying the PDE $\Delta w =g$, $g\in C(\overline {\mathbb {U}})$, $w(0)=0$. It is proved that $\mathcal {QC}(K,g)$ is a uniformly Lipschitz family. Moreover, if $|g|_\infty$ is small enough, then the family is uniformly bi-Lipschitz. The estimations are asymptotically sharp as $K \to 1$ and $|g|_\infty \to 0$, so $w\in \mathcal {QC}(K,g)$ behaves almost like a rotation for sufficiently small $K$ and $|g|_\infty$.