Locally invertible $\sigma$-harmonic mappings

TL;DR

This paper extends Lewy's classical theorem to planar $\sigma$-harmonic mappings, showing they are locally invertible under certain elliptic conditions, and also applies similar results to specific second order non-divergence equations.

Contribution

It introduces a generalization of Lewy's theorem to $\sigma$-harmonic mappings and related second order equations, establishing local invertibility in these contexts.

Findings

$\sigma$-harmonic mappings are locally invertible under elliptic conditions

Extension of Lewy's theorem to non-divergence equations

Applicable to pairs of solutions of certain second order equations

Abstract

We extend a classical theorem by H. Lewy to 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 $i=1,2$. A similar result is established for pairs of solutions of certain second order non--divergence equations.