Characterisation of electric fields in periodic composites

This is work done in collaboration with G.W. Milton and A. Treibergs (University of Utah). Our purpose is to characterise, among all the regular periodic gradient fields, the ones which are isotropically realisable electric fields, namely solutions of a conduction equation with a suitable isotropic conductivity. In any dimension a sufficient condition of realisability is that the gradient field does not vanish. This condition is also necessary in dimension two but not in dimension three. However, when the conductivity also needs to be periodic, the previous condition is shown to be not sufficient. Then, using the associated gradient flow a necessary and sufficient condition for the isotropic realisability in the torus is established and illustrated by several examples. The realisability of the matrix gradient fields and the less regular laminate fields is also investigated.