Sobolev Regularity for Monge-Ampère Type Equations

Abstract

Abstract. In this note we prove that, if the cost function satisfies some necessary structural conditions and the densities are bounded away from zero and infinity, then strictly c-convex potentials arising in optimal transportation belong to W 2,1+κ loc for some κ > 0. This generalizes some recents results [10,11,24] concerning the regularity of strictly convex Alexandrov solutions of the Monge-Ampère equation with right hand side bounded away from zero and infinity.

Untitled

“…As proved by the second author together with Kim and McCann [52], such a result is indeed true (see also [54,63,113]): Let us mention that further extensions of Theorem 3.3 to this general setting have been obtained in [37,81,82].…”

“…Note that Theorem 4.8 as well as the other results in [37,63,81,82,113] only deal with the interior regularity for optimal transport maps. It would be interesting to develop a boundary regularity theory, in the spirit of [20,22], for the class of equations (1.2) arising in optimal transport.…”

The Monge–Ampère equation and its link to optimal transportation

“…Then u ∈ C 1,α loc (X ) for any open set X ⊂ X where f is uniformly bounded away from zero. Let us mention that further extensions of Theorem 3.3 to this general setting have been obtained in [84,85,38].…”

“…Note that Theorem 4.8 as well as the other results in [84,85,38,66,117] only deal with the interior regularity for optimal transport maps. It would be interesting to develop a boundary regularity theory, in the spirit of [21,23], for the class of equations (1.2) arising in optimal transport.…”

On Asymptotic Behaviour and W 2, p Regularity of Potentials in Optimal Transportation

“…By a perturbation argument [37] and the local analysis in [20], the interior C 2 and C 2,α estimates were proved in [22], assuming respectively the Dini and Hölder continuity of f . Recently, the W 2,1+ε loc estimate for strictly c-convex potentials with the cost c satisfying (A3w) and the function f satisfying C −1 < f < C has been established [8] extending the corresponding estimate for strictly convex solutions of (1.5) [9,10,26]. See also [18,12,34] for related works on local geometry of transport costs and potentials, and [27,31,32,33] for more on optimal transportation.…”