Reconstruction of isotropic conductivities from non smooth electric fields

University of Rennes, INSA Rennes, CNRS, IRMAR – UMR 6625, 35000 Rennes, France

^* Corresponding author: mbriane@insa-rennes.fr

Received: 18 June 2017
Accepted: 9 February 2018

Abstract

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