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In this paper we study the BV regularity for solutions of variationalproblems in Optimal Transportation. As an application we recover BV estimatesfor solutions of some non-linear parabolic PDE by means of optimaltransportation techniques. We also prove that the Wasserstein projection of ameasure with BV density on the set of measures with density bounded by a givenBV 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 provethat the total variation of the projection does not exceed the total variationof the projected measure. This is an estimate which can be iterated, and istherefore very useful in some evolutionary PDEs (crowd motion,. . .). We alsoestablish some properties of the Wasserstein projection which are interestingin their own, and allow for instance to prove uniqueness of such a projectionin a very general framework.

BV Estimates in Optimal Transportation and Applications