Relaxed many-body optimal transport and related asymptotics

Optimization problems on probability measures in $\mathbb{R}^d$ 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 $c(x,y) =\ell(|x-y|)$ where $\ell: \mathbb{R}_+ \to [0,\infty]$ 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 \cite{bouchitte2020relaxed} and present a duality method which allows to compute the $\Gamma-$limit as $N\to\infty$ under very general assumptions on the cost $\ell(r)$. 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 $\ell_N(r)=\ell (r/\varepsilon)$ ($\varepsilon\ll 1$) and we show how the duality approach can be also used to determine the limit energy as $\varepsilon\to 0$ of a very large number $N_\varepsilon$ of particles.