Tobias Ried — Optimal Transportation, Monge-Ampère, and the Matching Problem

We present a fully variational approach to the regularity theory for the Monge-Ampère equation, or rather of optimal transportation, with interesting applications to the problem of optimally matching a realisation of a Poisson point process to the Lebesgue measure. Following De Giorgi’s strategy for the regularity theory of minimal surfaces, it is based on the approximation of the displacement by a harmonic gradient, and leads to a quantitative linearisation result for the Monge-Ampère equation. One of the benefits of our approach is that it also works for irregular data, in particular in situations where Caffarelli’s celebrated regularity theory is not expected to work.