  We analyze the combined effect of the long range Coulomb (LRC) interaction
and of surface energy on first order density-driven phase transitions in the
presence of a compensating rigid background. We study mixed states formed by
regions of one phase surrounded by the other in the case in which the scale of
the inhomogeneities is much larger than the interparticle distance. Two
geometries are studied in detail: spherical drops of one phase into the other
and a layered structure of one phase alternating with the other. We find the
optimum density profile in an approximation in which the free energy is a
functional of the local density (LDA). It is shown that an approximation in
which the density is assumed to be uniform (UDA) within each phase region gives
results very similar to those of the more involved LDA approach. Within the UDA
we derive the general equations for the chemical potential and the pressures of
each phase which generalize the Maxwell construction to this situation. The
equations are valid for a rather arbitrary geometry. We find that the
transition to the mixed state is quite abrupt i.e. inhomogeneities of the first
phase appear with a finite value of the radius and of the phase volume
fraction. The maximum size of the inhomogeneities is found to be on the scale
of a few electric field screening lengths. Contrary to the ordinary Maxwell
construction, the inverse specific volume of each phase depends here on the
global density in the coexistence region and can decrease as the global density
increases. The range of densities in which coexistence is observed shrinks as
the LRC interaction increases until it reduces to a singular point. We argue
that close to this singular point the system undergoes a lattice instability as
long as the inverse lattice compressibility is finite.
