Reconstruction of isotropic conductivities from non smooth electric fields

Briane, Marc

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 3, pp. 1173-1193.

In this paper we study the isotropic realizability of a given non smooth gradient field $\nabla u$ defined in ${\mathrm{R}}^d$, namely when one can reconstruct an isotropic conductivity $\sigma >0$ such that $\sigma \nabla u$ is divergence free in ${\mathrm{R}}^d$. On the one hand, in the case where $\nabla u$ is non-vanishing, uniformly continuous in ${\mathrm{R}}^d$ and $\Delta u$ is a bounded function in ${\mathrm{R}}^d$, we prove the isotropic realizability of $\nabla u$ using the associated gradient flow combined with the DiPerna, Lions approach for solving ordinary differential equations in suitable Sobolev spaces. On the other hand, in the case where $\nabla u$ is piecewise regular, we prove roughly speaking that the isotropic realizability holds if and only if the normal derivatives of $u$ on each side of the gradient discontinuity interfaces have the same sign. Some examples of conductivity reconstruction are given.

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