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 by figalli on 13 Aug 2024

[BibTeX]

Published Paper

Inserted: 24 feb 2012
Last Updated: 13 aug 2024

Journal: Anal. PDE
Year: 2013

ArXiv: 1202.5561 PDF

Abstract:

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


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