A variational approach to regularity theory for the Monge-Ampère equation

Michael Goldman
Univ. Paris 7
Thursday, May 3, 2018 - 16:00

In this talk I will present a new proof of the partial regularity of optimal transport maps. As opposed to the previous proof of Figalli and Kim which uses Caffarelli's approach to regularity of solutions of Monge-Ampère equations via maximum principle arguments, our proof is variational in nature. By using the fluid-dynamic formulation of optimal transportation (which usually goes by the name of Benamou-Brenier formulation) we prove that at every scale, the optimal transport map is close to the gradient of an harmonic function. This allows us to set up a Campanato iteration scheme to obtain the desired regularity. This is a joint work with F. Otto.

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