Reconstruction of isotropic conductivities from non smooth electric fields

TL;DR

This paper investigates conditions under which a non-smooth gradient field can be used to reconstruct an isotropic conductivity, providing theoretical results for both smooth and piecewise regular cases, with practical examples.

Contribution

It offers new criteria for isotropic conductivity reconstruction from non-smooth electric fields, extending previous work to less regular gradient fields.

Findings

Reconstruction possible when $ abla u$ is non-vanishing and uniformly continuous.

Reconstruction depends on the sign of normal derivatives across interfaces.

Examples demonstrate practical conductivity reconstructions.

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 $\triangle u$ 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.