Quantitative estimates on Jacobians for hybrid inverse problems

We consider σ-harmonic mappings, that is mappings U whose components ui solve a divergence structure elliptic equation div(σ∇ui) = 0, for i = 1, . . . , 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 eld of homogenization studying bounds for the e ective properties of composite materials.