Résumé : In this talk, we present a relaxation formula and duality theory for the multi-marginal Coulomb 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 interactions of k particles, with k lower than N. We then develop a duality framework and deduce primal-dual necessary and sufficient optimality conditions. Finaly we apply these results to a minimization problem involving a given continuous potential and we give evidence of a mass quantization effect for the optimal solutions.
This is a joint work with G. Bouchitté (Univ. Toulon), G. Buttazzo (Univ. Pisa) and L. De Pascale (Univ. Firenze)
Lieu : IMO, Salle 3L8
Département de Mathématiques
Bâtiment 307
Faculté des Sciences d'Orsay Université Paris-Sud F-91405 Orsay Cedex Tél. : +33 (0) 1-69-15-79-56 
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