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We propose a duality theory for multi-marginal repulsive cost that appears in optimal transport problems arising in Density Functional Theory. The related optimization problems involve probabilities on the entire space and, as minimizing sequences may lose mass at infinity, it is natural to expect relaxed solutions which are sub-probabilities. We first characterize the N-marginals relaxed cost in terms of a stratification formula which takes into account all k interactions with . We then develop a duality framework involving continuous functions vanishing at infinity and deduce primal-dual necessary and sufficient optimality conditions. Next we prove the existence and the regularity of an optimal dual potential under very mild assumptions. In the last part of the paper, we apply our results to a minimization problem involving a given continuous potential and we give evidence of a mass quantization effect for optimal solutions.
Keywords: Multi-marginal optimal transport, Duality and Relaxation, Coulomb cost, Quantization of minimizers
Bouchitté, Guy 1 ; Buttazzo, Giuseppe 2 ; Champion, Thierry 1 ; De Pascale, Luigi 3
@article{AIHPC_2021__38_1_61_0,
author = {Bouchitt\'e, Guy and Buttazzo, Giuseppe and Champion, Thierry and De Pascale, Luigi},
title = {Relaxed multi-marginal costs and quantization effects},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {61--90},
publisher = {Elsevier},
volume = {38},
number = {1},
year = {2021},
doi = {10.1016/j.anihpc.2020.06.004},
mrnumber = {4200477},
zbl = {1461.49046},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.06.004/}
}
TY - JOUR AU - Bouchitté, Guy AU - Buttazzo, Giuseppe AU - Champion, Thierry AU - De Pascale, Luigi TI - Relaxed multi-marginal costs and quantization effects JO - Annales de l'I.H.P. Analyse non linéaire PY - 2021 SP - 61 EP - 90 VL - 38 IS - 1 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.06.004/ DO - 10.1016/j.anihpc.2020.06.004 LA - en ID - AIHPC_2021__38_1_61_0 ER -
%0 Journal Article %A Bouchitté, Guy %A Buttazzo, Giuseppe %A Champion, Thierry %A De Pascale, Luigi %T Relaxed multi-marginal costs and quantization effects %J Annales de l'I.H.P. Analyse non linéaire %D 2021 %P 61-90 %V 38 %N 1 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.06.004/ %R 10.1016/j.anihpc.2020.06.004 %G en %F AIHPC_2021__38_1_61_0
Bouchitté, Guy; Buttazzo, Giuseppe; Champion, Thierry; De Pascale, Luigi. Relaxed multi-marginal costs and quantization effects. Annales de l'I.H.P. Analyse non linéaire, janvier – février 2021, Tome 38 (2021) no. 1, pp. 61-90. doi: 10.1016/j.anihpc.2020.06.004
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