Estimates for the dilatation of $σ$-harmonic mappings

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

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