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On a counter-example to quantitative Jacobian bounds. (Sur un contre-exemple aux bornes quantitatives du jacobien.) (English. French summary) Zbl 1327.35432
This note provides a counter-example to the local positivity of the Jacobian determinant for solutions of the conductivity equation in dimension 3. It shows that the sign of the determinant cannot be imposed by an a priori choice of boundary data in \(H^{1/2}(\partial\Omega)\) depending only on the upper and lower bound of the conductivity, even locally.

MSC:
35R30 Inverse problems for PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
78M40 Homogenization in optics and electromagnetic theory
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J57 Boundary value problems for second-order elliptic systems
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