Geodesic
Quantitative estimates on Jacobians for hybrid inverse problems
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 8 (2015) no. 3, pp. 25-41

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We consider $\sigma$-harmonic mappings, that is mappings $U$ whose components $u_i$ solve a divergence structure elliptic equation ${\rm div} (\sigma \nabla u_i)=0$, 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.
Keywords: elliptic equations; Beltrami operators; hybrid inverse problems; composite materials. 
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     title = {Quantitative estimates on {Jacobians} for hybrid inverse problems},
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G. Alessandrini; V. Nesi. Quantitative estimates on Jacobians for hybrid inverse problems. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 8 (2015) no. 3, pp. 25-41. http://geodesic.mathdoc.fr/item/VYURU_2015_8_3_a1/