*Accepted Paper*

**Inserted:** 31 oct 2014

**Last Updated:** 25 jan 2016

**Journal:** Arch. Ration. Mech. An.

**Year:** 2015

**Abstract:**

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,\dots). 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.

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