Isotropic Realizability of Current Fields in ℝ3

This paper deals with the isotropic realizability of a given regular divergence free field j in R3 as a current field, namely to know when j can be written as σ∇u for some isotropic conductivity σ > 0, and some gradient field ∇u. The local isotropic realizability in R3 is obtained by Frobenius’ theorem provided that j and curl j are orthogonal in R3. A counter-example shows that Frobenius’ condition is not sufficient to derive the global isotropic realizability in R3. However, assuming that (j, curl j, j × curl j) is an orthogonal basis of R3, an admissible conductivity σ is constructed from a combination of the three dynamical flows along the directions j/|j|, curl j/|curl j| and (j/|j|2) × 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)×curl j. Several examples illustrate the sharpness of the realizability conditions.