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Frank, Rupert L. and Lieb, Elliott H. (2018): A "Liquid-Solid" Phase Transition in a Simple Model for Swarming, Based on the "No Flat-Spots" Theorem for Subharmonic Functions. In: Indiana University Mathematics Journal, Vol. 67, No. 4: pp. 1547-1569

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We consider a family of non-local shape optimization problems, which are motivated by a simple model for swarming and other self-assembly/aggregation models, and prove the existence of different phases for several of them. A technical key ingredient, which we establish, is that a strictly subharmonic function cannot be constant on a set of positive measure.

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