A variational regularity theory for optimal transportation, and application to the matching problem.

Date: 23 June 2025 - 27 June 2025

Abstract

In this mini-course, we shall explain the variational approach to regularity theory for optimal transportation. As a text material, we refer to the lecture notes arXiv:2303.14462. As opposed to the maximum-principle based regularity theory by Caffarelli for the Monge-Ampère equation, which has been extended to $\varepsilon$-regularity by Figalli et al, this approach follows a strategy analogous to minimal surfaces, in the sense that at its core is a harmonic approximation. The advantage of this approach lies in its robustness, which allows to apply it to the optimal matching problem. For an introduction of the latter, see recorded-lectures under “From combinatorics to partial differential equations”. This connects to work of Parisi et al and Ambrosio et al.