Screening effects in Coulomb frustrated phase separation

We solve a model of phase separation among two competing phases frustrated by the long-range Coulomb interaction in two and three dimensions (2D/3D) taking into account finite compressibility effects. In the limit of strong frustration in 2D, we recover the results of R. Jamei, S. Kivelson, and B. Spivak, Phys. Rev. Lett. 94, 056805 (2005) and the system always breaks into domains in a narrow range of densities, no matter how big is the frustration. For weak frustration in 2D and for arbitrary frustration in 3D the finite compressibility of the phases is shown to play a fundamental role. Our results clarify the different role of screening in 2D and 3D systems. We discuss the thermodynamic stability of the system near the transition to the phase separated state and the possibility to observe it in real systems.