Quasiconformal mappings and H\"older continuity

TL;DR

This paper proves that certain quasiconformal mappings are H"older continuous under conditions on their Laplacian, extending regularity results based on integrability conditions of the Laplacian.

Contribution

It establishes H"older continuity of quasiconformal mappings with Laplacian in L^p, generalizing previous regularity results for these mappings.

Findings

H"older continuity holds for mappings with Laplacian in L^p, for p in (n/2, n)

Explicit H"older exponent depends on p and dimension n

Mappings with Laplacian in L^n are H"older continuous for all 0<α<1

Abstract

We establish that every $K$-quasiconformal mapping $w$ of the unit ball $\IB$ onto a $C^2$-Jordan domain $\Omega$ is H\"older continuous with constant $\alpha= 2-\frac{n}{p}$, provided that its weak Laplacean $\Delta w$ is in $ L^p(\IB)$ for some $n/2<p<n$. In particular it is H\"older continuous for every $0<\alpha<1$ provided that $\Delta w\in L^n(\IB)$.