Relaxed multi-marginal costs and quantization effects

Guy Bouchitté (1) , Giuseppe Buttazzo (2) , Thierry Champion (1) , Luigi de Pascale (3)

1 IMATH - Institut de Mathématiques de Toulon - EA 2134
2 Dipartimento di Matematica [Pisa]
3 DiMal - Dipartimento di Matematica e Informatica [Firenze]

We propose a duality theory for multi-marginal repulsive cost that appear 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 k ≤ N. 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.

Mots clés: Multi-marginal optimal transport; Duality and Relaxation; Coulomb cost; Quantization of minimizers 2010 Mathematics Subject Classification: 49J45; 49N15; 49K30; 35Q40

Domaines: Optimisation et contrôle [math.OC]; Physique mathématique [math-ph]

Guy Bouchitté, Giuseppe Buttazzo, Thierry Champion, Luigi de Pascale. Relaxed multi-marginal costs and quantization effects. 2019. ⟨hal-02352469⟩