  In this paper, we find a class of special inclusions that have the same
property with respect to second order linear partial differential equations as
holds for ellipsoids. That is, in the simplest case and in physical terms,
constant magnetization of the inclusion implies constant magnetic field on the
inclusion. The special inclusions are found as solutions of a simple
variational inequality. This variational inequality allows us to prescribe the
connectivity and periodicity properties of the inclusions. For example we find
periodic arrays of inclusions in two and three dimensions for which constant
magnetization of the inclusions implies constant magnetic field on the
inclusions. The volume fraction of the inclusions can be any number between
zero and one. We find such inclusions with any finite number of components and
components that are multiply connected. These special inclusions enjoy many
useful properties with respect to homogenization and energy minimization. For
example, we use them to give new results on a) the effective properties of
two-phase composites and b) optimal bounds and optimal microstructures for
two-phase composites.
