Relaxed many-body optimal transport and related asymptotics.

Optimization problems on probability measures in ℝ d are considered where the cost functional involves multi-marginal optimal transport. In a model of N interacting particles, for example 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 [4] 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 ℓ N ⁢ (r) = ℓ ⁢ (r / ε) ( ε ≪ 1 ) and we show how the duality approach can also be used to determine the limit energy as ε → 0 of a very large number N ε of particles. [ABSTRACT FROM AUTHOR]