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.

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

#### 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11368/2830483
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