Title: Optimal transportation and Monge-Ampere type equations

Speaker: Jiakun Liu(Princeton University)

Time: July 21, 2011, 13:30-15:30 pm

Venue: Room 1218 at BICMR, Resource Plaza, Peking University

Abstract: In the first part, we introduce some background of the optimal transportation problem.  Under appropriate conditions on the cost function, the optimal mapping is uniquely determined by the potential function, which satisfies a Monge-Ampere type equation. Therefore, in order to study the regularity of optimal mapping, it suffices to study the regularity of solutions to the Monge-Ampere equation. In the second part, we present some recent regularity results for the Monge-Ampere equations arising from optimal transportation. This is a joint work with Neil Trudinger and Xu-Jia Wang. We obtain the Holder and more general continuity estimates for second derivatives of solutions, when the inhomogeneous term is merely Holder or Dini continuous. We would also like to show some interesting examples of cost function, and applications in conformal geometry.