2022/10/12 by Bindini, Ugo, Bouchitté, Guy · 1 citation
#49J45 #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2210.06532
Optimization problems on probability measures in ℝd are considered where the cost functional involves multi-marginal optimal transport. In a model of N interacting particles, like in Density Functional Theory, the interaction cost is repulsive and described by a two-point function c(x,y) =ℓ(|x-y|) where ℓ: ℝ+ → [0,∞] is decreasing to zero at infinity. Due to a possible loss of mass at infinity, non existence may occur and relaxing the initial problem over sub-probabilities becomes necessary. In this paper we characterize the relaxed functional generalizing the results of \citebouchitte2020relaxed and present a duality method which allows to compute the Γ-limit as N→∞ under very general assumptions on the cost ℓ(r). We show that this limit coincides with the convex hull of the so-called direct energy. Then we study the limit optimization problem when a continuous external potential is applied. Conditions are given with explicit examples under which minimizers are probabilities or have a mass <1 . In a last part we study the case of a small range interaction ℓN(r)=ℓ (r/ε) (ε≪ 1) and we show how the duality approach can be also used to determine the limit energy as ε→ 0 of a very large number Nε of particles.