Guido De Philippis, Alessio Figalli
Published 2012-02-24, updated 2012-11-16Version 3
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.
Pointwise $C^{2,α}$ estimates at the boundary for the Monge-Ampere equation
Partial Regularity for optimal transport maps
Global $W^{2,p}$ estimates for the Monge-Ampere equation
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