Isotropic Realizability of Current Fields in $\mathbb{R}^3$

Abstract

This paper deals with the isotropic realizability of a given regular divergence free field $j$ in $\mathbb{R}^3$ as a current field, namely, to know when $j$ can be written as $\sigma\nabla u$ for some isotropic conductivity $\sigma>0$ and some gradient field $\nabla u$. The local isotropic realizability in $\mathbb{R}^3$ is obtained by Frobenius' theorem provided that $j$ and $\mbox{curl}\,j$ are orthogonal in $\mathbb{R}^3$. A counterexample shows that Frobenius' condition is not sufficient to derive the global isotropic realizability in $\mathbb{R}^3$. However, assuming that $(j,\mbox{curl}\,j,j\times\mbox{curl}\,j)$ is an orthogonal basis of $\mathbb{R}^3$, an admissible conductivity $\sigma$ is constructed from a combination of the three dynamical flows along the directions $j/|j|$, $\mbox{curl}\,j/|\mbox{curl}\,j|,$ and $(j/|j|^2)\times\mbox{curl}\,j$. When the field $j$ is periodic, the isotropic realizability in the torus needs in addition a boundedness assumption satisfied by the flow along the t...