  We derive the phase diagram of a paradigmatic model of Coulomb frustrated
phase separation in two-dimensional systems with negative short-range
electronic compressibility. We consider the system subject either to the truly
three-dimensional long-range Coulomb interaction (LRC) and to a two-dimensional
LRC with logarithmic-like behavior. In both cases we find that the transition
from the homogeneous phase to the inhomogeneous phase is generically
first-order except for a critical point. Close to the critical point,
inhomogeneities arrange in a triangular lattice with a subsequent first-order
topological transition to stripe-like objects by lowering the Coulomb
frustration. A proliferation of inhomogeneities which have inside smaller
inhomogeneities is expected near all the transition lines in systems embedded
in the three-dimensional LRC alone.
