I will discuss a problem of energy-driven pattern formation, in which the appearance of two distinct phases caused by short-range attractive forces is frustrated by a long-range repulsive force. I will focus on the regime of strong compositional asymmetry, in which one of the phases has very small volume fraction, thus creating small “droplets” of the minority phase in a “sea” of the majority phase. I will present a setting for the study of Gamma-convergence of the governing energy functional in the regime leading to many droplets. The Gamma-limit and the properties of almost minimizers with prescribed limit density will then be established in the important physical case when the long-range repulsive force is Coulombic in two space dimensions. This is joint work with D. Goldman and S. Serfaty.
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