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Quantitative estimates on Jacobians for hybrid inverse problems

We consider $σ$-harmonic mappings, that is mappings $U$ whose components $u_i$ solve a divergence structure elliptic equation ${\rm div} (σ\nabla u_i)=0$, for $i=1,\ldots,n $. We investigate whether, with suitably prescribed Dirichlet data, the Jacobian determinant can be bounded away from zero. Results of this sort are required in the treatment of the so-called hybrid inverse problems, and also in the field of homogenization studying bounds for the effective properties of composite materials.