Calculus of Variations and Geometric Measure Theory

G. De Philippis - A. Figalli

Second order stability for the Monge-Ampere equation and strong Sobolev convergence of optimal transport maps

created by dephilipp on 24 Feb 2012
modified on 30 Oct 2017


Accepted Paper

Inserted: 24 feb 2012
Last Updated: 30 oct 2017

Journal: Anal. PDE
Year: 2012

ArXiv: 1202.5561 PDF


The aim of this note is to show that Alexandrov solutions of the Monge-Ampere equation, with right hand side bounded away from zero and infinity, converge strongly in $W^{2,1}_{loc}$ if their right hand side converge strongly in $L^1_{loc}$. As a corollary we deduce strong $W^{1,1}_{loc}$ stability of optimal transport maps.

Keywords: optimal transportation, stability, Monge-Ampère equation