Weekly Seminar
A variational approach to the regularity theory for optimal transportation
Felix Otto (MPI-MIS, Leipzig (Germany))
Thu, 19 Aug 2021 • 10:30-11:30h • Templergraben 55, lecture hall II (max. 33 persons, so please announce participation via email)

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-Ampère 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 $C^{1,\alpha}$-level for the optimal map, capitalizing on affine invariance.

On the one hand, this allows to re-prove the $\epsilon$-regularity result (Figalli-Kim, De Philippis-Figalli) bypassing Caffarelli’s celebrated theory. This also extends to general cost functions, which is joint work with M. Prodhomme and T. Ried.

On the other hand, due to its robustness and low-regularity approach, it can be used to study the popular problem of matching two independent Poisson point processes. For example, it can be used to prove non-existence of a stationary cyclically monotone coupling, which is joint work with M. Huesmann and F. Mattesini.