Fonctionne avec Indico
v3.2.9
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 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 optimal 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 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. This variational approach allows to re-prove the $\epsilon$-regularity result (Figalli-Kim, De Philippis-Figalli) bypassing Caffarelli's theory.
However, the advantage of the variational approach resides in its robustness regarding the regularity of the measures, which can be arbitrary measures, and in terms of the problem formulation, e. g. by its extension to almost minimizers (with M. Prod'homme and T. Ried). The former for instance is crucial in order to tackle the widely popular matching problem (with F. Mattesini and M. Huesmann). The latter is convenient when treating more general than square Euclidean cost functions.