BV Estimates in Optimal Transportation and Applications

Guido de Philippis (1) , Alpár Richard Mészáros (2) , Filippo Santambrogio (2) , Bozhidar Velichkov (3)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 CVGI - Calcul des Variations, Géométrie, Image

In this paper we study the BV regularity for solutions of variational problems in Optimal Transportation. As an application we recover BV estimates for solutions of some non-linear parabolic PDE by means of optimal transportation techniques. We also prove that the Wasserstein projection of a measure with BV density on the set of measures with density bounded by a given BV function f is of bounded variation as well. In particular, in the case f = 1 (projection onto a set of densities with an L^\infty bound) we precisely prove that the total variation of the projection does not exceed the total variation of the projected measure. This is an estimate which can be iterated, and is therefore very useful in some evolutionary PDEs (crowd motion,. . .). We also establish some properties of the Wasserstein projection which are interesting in their own, and allow for instance to prove uniqueness of such a projection in a very general framework.

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Guido de Philippis, Alpár Richard Mészáros, Filippo Santambrogio, Bozhidar Velichkov. BV Estimates in Optimal Transportation and Applications. Archive for Rational Mechanics and Analysis, 2016, 219 (2), pp.829-860. ⟨10.1007/s00205-015-0909-3⟩. ⟨hal-01079336v2⟩