On quasiconformal selfmappings of the unit disk and elliptic PDE in the plane

TL;DR

This paper proves that quasiconformal self-mappings of the unit disk satisfying a certain elliptic PDE inequality are Lipschitz continuous, extending previous results that only considered the Laplace operator.

Contribution

It extends existing results by establishing Lipschitz continuity for quasiconformal mappings under a broader class of elliptic PDE inequalities.

Findings

Quasiconformal mappings satisfying the PDE inequality are Lipschitz continuous.

The result generalizes previous work limited to the Laplace operator.

Provides conditions under which quasiconformal maps are Lipschitz.

Abstract

We prove the following theorem: if $w$ is a quasiconformal mapping of the unit disk onto itself satisfying elliptic partial differential inequality $|L[w]|\le \mathcal{B}|\nabla w|^2+\Gamma$, then $w$ is Lipschitz continuous. This {result} extends some recent results, where instead of an elliptic differential operator is {only} considered {the} Laplace operator.