BV estimates in optimal transportation and applications

I will study BV regularity for solutions of variational problems in Optimal Transportation and present some applications. In particular I will focus on $BV$ bounds for the Wasserstein projection on the set of measure with density bounded by a prescribed $B{V}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ function $f$. I will also show how to recover BV estimates for solutions of some non-linear parabolic PDE by means of optimal transportation techniques. This is a joint work with A. Meszaros, F. Santambrogio and B. Velichkov.