From radial symmetry to fractal behavior of aggregation equilibria for repulsive–attractive potentials
José A. Carrillo, Ruiwen Shu
Abstract
Abstract For the interaction energy with repulsive–attractive potentials, we give generic conditions which guarantee the radial symmetry of the local minimizers in the infinite Wasserstein distance. As a consequence, we obtain the uniqueness of local minimizers in this topology for a class of interaction potentials. We introduce a novel notion of concavity of the interaction potential allowing us to show certain fractal-like behavior of the local minimizers. We provide a family of interaction potentials such that the support of the associated local minimizers has no isolated points and any superlevel set has no interior points.
Topics & Concepts
UniquenessMathematicsSymmetry in biologySymmetry (geometry)FractalClass (philosophy)Set (abstract data type)Pure mathematicsStatistical physicsMathematical analysisTopology (electrical circuits)CombinatoricsPhysicsGeometryComputer scienceProgramming languageArtificial intelligenceGeometric Analysis and Curvature FlowsTheoretical and Computational PhysicsNonlinear Partial Differential Equations