Isotropic realizability of current fields in R³

2014/09/26 by Marc Briane, Graeme W. Milton, Briane, Marc +1
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP

paper · pdf · doi:10.48550/arxiv.1409.7658

22 pages

arxiv created 2014/09/26 · arxiv updated 2014/09/29

Abstract

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 sigma Du for some isotropic conductivity sigma, and some gradient field Du. 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 x curl j) is an orthogonal basis of R³, 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.

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