Sobolev Regularity for Monge-Ampère Type Equations

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 $W^{2,1+\kappa}_{\rm loc}$ for some $\kappa>0$. This generalizes some recents results concerning the regularity of strictly convex Alexandrov solutions of the Monge--Ampère equation with right-hand side bounded away from zero and infinity [G. De Philippis and A. Figalli, Invent. Math., 192 (2013), pp. 55--69; G. De Philippis, A. Figalli, and O. Savin, Math. Ann., to appear; T. Schmidt, Adv. Math., to appear].