An exterior optimal transport problem

Un problème de transport optimal extérieur

Jules Candau-Tilh (1, 2) , Michael Goldman (3, 4) , Benoît Merlet (1, 2)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 RAPSODI - Reliable numerical approximations of dissipative systems
3 CMAP - Centre de Mathématiques Appliquées de l'Ecole polytechnique
4 CNRS - Centre National de la Recherche Scientifique

This paper deals with a variant of the optimal transportation problem. Given f ∈ L 1 (R d , [0, 1]) and a cost function c ∈ C(R d × R d) of the form c(x, y) = k(y − x), we minimise ∫ c dγ among transport plans γ whose first marginal is f and whose second marginal is not prescribed but constrained to be smaller than 1 − f. Denoting by Υ(f) the infimum of this problem, we then consider the maximisation problem sup{Υ(f) : ∫ f = m} where m > 0 is given. We prove that maximisers exist under general assumptions on k, and that for k radial, increasing and coercive these maximisers are the characteristic functions of the balls of volume m.

Mots clés: Optimal transport; Dual problem; Existence of maximisers

Domaines: Equations aux dérivées partielles [math.AP]

Jules Candau-Tilh, Michael Goldman, Benoît Merlet. An exterior optimal transport problem. Calculus of Variations and Partial Differential Equations, inPress. ⟨hal-04196005v2⟩