Reconstruction of isotropic conductivities from non smooth electric fields

Abstract

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 $△u$ 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.

ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, A Paraître