Quantitative estimates on Jacobians for hybrid inverse problems

Abstract

We consider $\sigma$-harmonic mappings, that is mappings $U$ whose components $u_i$ solve a divergence structure elliptic equation $\mathrm{div}(\sigma \nabla u_i)=0$, for $i=1,\dots ,n$. 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.

15 pages, submitted

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