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- W4320484906 abstract "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 $Ntoinfty$ 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 $ell_N(r)=ell (r/varepsilon)$ ($varepsilonll 1$) and we show how the duality approach can be also used to determine the limit energy as $varepsilonto 0$ of a very large number $N_varepsilon$ of particles." @default.
- W4320484906 created "2023-02-14" @default.
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- W4320484906 date "2022-10-12" @default.
- W4320484906 modified "2023-09-28" @default.
- W4320484906 title "Relaxed many-body optimal transport and related asymptotics" @default.
- W4320484906 doi "https://doi.org/10.48550/arxiv.2210.06532" @default.
- W4320484906 hasPublicationYear "2022" @default.
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