Isotropic realizability of current fields in R^3

TL;DR

This paper investigates conditions under which a divergence-free current field in three-dimensional space can be realized as an isotropic conductivity times a gradient field, providing both local and global criteria, and constructing explicit conductivities.

Contribution

It establishes Frobenius' theorem for local realizability, identifies conditions for global realizability, and constructs explicit conductivities using dynamical flows in R^3.

Findings

Frobenius' condition ensures local realizability when j and curl j are orthogonal.

Counter-example shows Frobenius' condition alone is insufficient for global realizability.

Explicit construction of admissible conductivity sigma using flows along specific directions.

Abstract

This paper deals with the isotropic realizability of a given regular divergence free field j in R^3 as a current field, namely to know when j can be written as sigma Du for some isotropic conductivity sigma, and some gradient field Du. The local isotropic realizability in R^3 is obtained by Frobenius' theorem provided that j and curl j are orthogonal in R^3. A counter-example shows that Frobenius' condition is not sufficient to derive the global isotropic realizability in R^3. However, assuming that (j, curl j, j x curl j) is an orthogonal basis of R^3, an admissible conductivity sigma is constructed from a combination of the three dynamical flows along the directions j/|j|, curl j/|curl j| and (j/|j|^2) x 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 third direction (j/|j|^2) x \curl j. Several examples illustrate the sharpness of the realizability conditions.