  We review the use of Legendre transforms in the formulation of electrostatic
energies in condensed matter. We show how to render standard functionals
expressed in terms of the electrostatic potential, Phi, convex - at the cost of
expressing them in terms of the vector field D. This leads to great
simplification in the formulation of numerical minimisation of electrostatic
energies coupled to other physical degrees of freedom. We also demonstrate the
equivalence of recent functionals for dielectrics derived using field theory
methods to classical formulations in terms of the electric polarisation.

Statistical Mechanics