Journée de rentrée 2023 du GT CalVa

Minimizing non local functionals on measures. Relaxation and asymptotics.

16 janv. 2023, 11:00

1h

Amphi Turing (Université Paris-Cité (campus Grands Moulins))

Orateur

Guy Bouchitté (IMATH, Université de Toulon)

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 $\ell :{\mathbb{R}}_{+}\to [0,\mathrm{\infty }]$ 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 $\mathrm{\Gamma }-$limit as $N\to \mathrm{\infty }$. 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 ${\mathbb{R}}^d$.