Cardinality estimation of support of the global minimizer for the interaction energy with mildly repulsive potentials

Abstract

Abstract In this work, we give a cardinality estimation of the support of any global minimizer for the interaction energy with mildly repulsive potentials. For any global minimizer μ , the fact that supp ( μ ) consists of finitely many points was proved in Carrillo et al. (2017). Moreover, the authors found a cardinality estimation of the support of μ on R 1 . In order to advance some improvement of this result, we give an upper estimate of card ( supp ( μ ) ) in arbitrary finite dimension.