A variational approach to the regularity theory for the Monge-Ampère equation
We present a purely variational approach to the regularity theory for the Monge-Ampère equation, or rather optimal transportation, introduced with M. Goldman. 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, which leads to a One-Step Improvement Lemma, and feeds into a Campanato iteration on the C1,α-level for the displacement, capitalizing on affine invariance. On the one hand, this allows to reprove the C1,α-regularity result (Figalli-Kim, De Philippis-Figalli) bypassing Caffarelli’s celebrated theory. This also extends to boundary regularity (Chen-Figalli), which is joint work with T. Miura. On the other hand, due to its robustness, it can be used as a large-scale regularity theory for the problem of matching the Lebesgue measure to the Poisson measure in the thermodynamic limit.
This is joint work with M. Goldman and M. Huesmann.