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

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 $W_{loc}^{2,1}$ if their right hand side converge strongly in $L_{loc}^1$. As a corollary we deduce strong $W_{loc}^{1,1}$ stability of optimal transport maps.