A variational approach to boundary regularity of optimal transport maps

One of the main topics in optimal transport theory is the regularity of optimal transport maps. As is classically observed by Caffarelli, the regularity of maps is closely related to the geometry of densities; in particular, for general densities (with possibly-nonconvex support) the corresponding map is not necessarily fully smooth and one can expect only partial or conditional regularity. In this talk I will first briefly review previous regularity results and then present a recent joint work with Felix Otto on a variational boundary $\varepsilon$-regularity theory.