The 2021 DiPerna Lecture will be given by Felix Otto (Max Planck Institute) on December 3, 2021, 4PM in 2 Physics
Title: A Variational Regularity Theory for Optimal Transportation
Abstract: The optimal transportation of one measure into another, leading to the notion of their Wasserstein distance, is a problem in the calculus of variations with a wide range of applications. The regularity theory for the optimal map is subtle and was revolutionized by Caffarelli. This approach relies on the fact that the Euler-Lagrange equation of this variational problem is given by the Monge-Ampére equation. The latter is a prime example of a fully nonlinear (degenerate) elliptic equation, amenable to comparison principle arguments.
We present a purely variational approach to the regularity theory for optimal transportation, introduced with M. Goldman and refined with M. Huesmann. Following De Giorgi's philosophy for the regularity theory of minimal surfaces, it is based on the approximation of the displacement by a harmonic gradient, through the construction of a variational competitor. This leads to a ``one-step improvement lemma'', and feeds into a Campanato iteration on the C1,α-level for the optimal map, capitalizing on affine invariance.