A `liquid-solid' phase transition in a simple model for swarming, based on the `no flat-spots' theorem for subharmonic functions

TL;DR

This paper investigates a non-local shape optimization model inspired by swarming behavior, proving the existence of different phases and establishing a key mathematical property related to subharmonic functions.

Contribution

It introduces a novel phase transition concept in a simple swarming model and proves a new mathematical property of subharmonic functions relevant to the model.

Findings

Existence of multiple phases in the model

Proof that strictly subharmonic functions cannot be constant on sets of positive measure

Mathematical validation of phase transition in swarming models

Abstract

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