Calculus of Variations and Geometric Measure Theory

G. De Philippis - A. Figalli

Sobolev regularity for Monge-Ampère type equations

created by dephilipp on 10 Nov 2012
modified on 30 Oct 2017


Accepted Paper

Inserted: 10 nov 2012
Last Updated: 30 oct 2017

Journal: SIAM J. Math. Anal.
Year: 2012

ArXiv: 1211.2341 PDF


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-Amp\`ere equation with right hand side bounded away from zero and infinity.