Calculus of Variations and Geometric Measure Theory
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G. De Philippis - A. Figalli

Sobolev regularity for Monge-Ampère type equations

created by dephilipp on 10 Nov 2012
modified on 18 Apr 2013


Accepted Paper

Inserted: 10 nov 2012
Last Updated: 18 apr 2013

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


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 [9], [10], [22] concerning the regularity of strictly convex Alexandrov solutions of the Monge-Ampère equation with right hand side bounded away from zero and infinity.


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