Isotropic realizability of current fields in $R^3$

Marc Briane (1) , Graeme W. Milton (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 Department of Mathematics - University of Utah

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.

Mots clés: current field; isotropic conductivity; Frobenius' condition; dynamical flow

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Marc Briane, Graeme W. Milton. Isotropic realizability of current fields in $R^3$. SIAM Journal on Applied Dynamical Systems, 2015, 14 (2), pp.1165-1188. ⟨10.1137/140989121⟩. ⟨hal-01069082⟩