Estimates for the dilatation of $\sigma$-harmonic mappings

TL;DR

This paper investigates conditions under which planar $\sigma$-harmonic mappings are quasiconformal, providing quantitative bounds that link the properties of the coefficient matrix $\sigma$ to the mapping's geometric distortion.

Contribution

The authors establish new quantitative bounds showing that locally invertible $\sigma$-harmonic mappings are quasiconformal under mild regularity assumptions involving $ ext{det}\sigma$ and its antisymmetric part.

Findings

Proved that $\sigma$-harmonic mappings are quasiconformal under certain conditions.

Derived bounds relating $ ext{det}\sigma$ and antisymmetric parts to quasiconformality.

Extended understanding of geometric properties of solutions to divergence elliptic equations.

Abstract

We consider planar $\sigma$-harmonic mappings, that is mappings $U$ whose components $u^1$ and $u^2$ solve a divergence structure elliptic equation ${\rm div} (\sigma \nabla u^i)=0$, for $i=1,2$. We investigate whether a locally invertible $ \sigma$-harmonic mapping $U$ is also quasiconformal. Under mild regularity assumptions, only involving $\det \sigma$ and the antisymmetric part of $\sigma$, we prove quantitative bounds which imply quasiconformality.