  In this paper we study the realizability of a given smooth periodic gradient
field $\nabla u$ defined in $\RR^d$, in the sense of finding when one can
obtain a matrix conductivity $\si$ such that $\si\nabla u$ is a divergence free
current field. The construction is shown to be always possible locally in
$\RR^d$ provided that $\nabla u$ is non-vanishing. This condition is also
necessary in dimension two but not in dimension three. In fact the
realizability may fail for non-regular gradient fields, and in general the
conductivity cannot be both periodic and isotropic. However, using a dynamical
systems approach the isotropic realizability is proved to hold in the whole
space (without periodicity) under the assumption that the gradient does not
vanish anywhere. Moreover, a sharp condition is obtained to ensure the
isotropic realizability in the torus. The realizability of a matrix field is
also investigated both in the periodic case and in the laminate case. In this
context the sign of the matrix field determinant plays an essential role
according to the space dimension.
