Locally invertible $\sigma$-harmonic mappings

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 $\mathrm{d}\mathrm{i}\mathrm{v}(\sigma \mathrm{\nabla }{u}^i)=0$ , for $i=1,2$. A similar result is established for pairs of solutions of certain second order non–divergence equations.