Reconstruction of isotropic conductivities from non smooth electric fields

In this paper we study the isotropic realizability of a given non smooth gradient field $\nabla u$ defined in $\mathbb{R}^d$, namely when one can reconstruct an isotropic conductivity $\sigma>0$ such that $\sigma\nabla u$ is divergence free in $\mathbb{R}^d$. On the one hand, in the case where $\nabla u$ is non-vanishing, uniformly continuous in $\mathbb{R}^d$ and $\triangle u$ is a bounded function in $\mathbb{R}^d$, 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.