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

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

created by dephilipp on 24 Feb 2012
modified by figalli on 15 Nov 2012


Accepted Paper

Inserted: 24 feb 2012
Last Updated: 15 nov 2012

Journal: Anal. PDE
Year: 2012


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

Keywords: optimal transportation, stability, Monge-Ampère equation


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