Orateur
Description
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, 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 talk we will describe the relaxed functional to be minimized as well as its $\Gamma-$limit as $N\to\infty$. 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 consider the case of a infinitesimal range interaction cost $\ell_N(r)=\ell (r/e)$ ($e\ll 1$) with the aim of determining the mean-field limit energy as $e\to 0$ of a very large number $N_e$ of particles confined in a given compact subset of $\mathbb{R}^d$.
