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This Colloquium is part of the event GSSI MeMocs Tullio Levi-Civita day, starting at 3:00pm. For more information about the event and the International Prize "Tullio Levi-Civita" for the Mathematical and Mechanical Sciences, please consult the following webpage: https://indico.gssi.it/event/371/
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 statistical interpretation. The regularity theory for the coupling 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 opti- mal 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 variational approach allows to re-prove the ε-regularity result of Figalli et. al. bypassing Caffarelli’s theory. The advantage of the variational approach lies in its robustness regarding the regularity of the measures, which can be arbitrary measures. In particular, it can be applied to the optimal matching between the two empirical mea- sures, as formulated by Ajtai et. al.
The connection to the Monge-Ampère equation and ultimately to the Poisson equation, enabled Parisi et. al. to give a finer characterization, made rigorous by Ambrosio et. al. on the macroscopic level. As one application of our variational approach we can go down to the mesoscopic level and obtain a sharp non-existence result in the critical 2-dimensional case (work with M. Huesmann and F. Mattesini).