Second order stability for the Monge-Ampere equation and strong Sobolev   convergence of optimal transport maps
  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. 
Guido De Philippis, Alessio Figalli
math.AP