*Accepted Paper*

**Inserted:** 31 oct 2014

**Last Updated:** 18 jan 2018

**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,. . .). 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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