View AbstractAbstract: In this talk the main question that I will consider is the regularity of solutions of certain variational problems in optimal transport. In particular I will be interested in the Wasserstein projection of a measure with BV density on the set of measures with densities bounded by a given BV function f. I will show that the projected measure is of bounded variation as well with a precise estimate of its BV norm. Of particular interest is the case f = 1, corresponding to a projection onto a set of densities with an $L^\infty$ bound, where one can prove that the total variation decreases by the projection. This estimate and, in particular, its iterations have a natural application to some evolutionary PDEs as, for example, the ones describing a crowd motion. In fact, as an application of our results, one can obtain BV estimates for solutions of some non-linear parabolic PDEs by means of optimal transport techniques. The talk is based on a joint work with G. De Philippis (SISSA, Italy), F. Santambrogio (Orsay, France) and B. Velichkov (Grenoble, France).