Felix Otto

Max Planck Institute for Mathematics

A variational approach to the regularity theory for optimal transportation

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 pioneered by Caffarelli. That 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. 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, which 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 reprove the \(\epsilon\)-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, and to general cost functions, which is joint work with M. Prodhomme and T. Ried.

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.