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

Abstract

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