Minimizers for a one-dimensional interaction energy

Abstract

We solve explicitly a certain minimization problem for probability measures in one dimension involving an interaction energy that arises in the modelling of aggregation phenomena. We show that in a certain regime minimizers are absolutely continuous with an unbounded density, thereby settling a question that was left open in previous works.Recently, Davies, Lim and McCann [10, Theorem 2.2] have shown that for α ≥ 3 the minimizers for E α are precisely of the form µ = 2 −1 (δ a−1/2 + δ a+1/2 ) for some a ∈ R. Earlier, Kang, Kim, Lim and Seo [15, Theorem 2] had shown that in the case 2 < α < 3, for any m ∈ (0, 1) and a ∈ R the measure mδ a−1/2 + (1 − m)δ a+1/2 is a saddle point for E α (with respect to the ∞-Wasserstein metric) and therefore, in