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

Alessio Figalli; Guido De Philippis

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

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arxiv 1202.5561 v3 pith:DAAHRSVI submitted 2012-02-24 math.AP

Guido De Philippis , Alessio Figalli This is my paper

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keywords convergeequationhandmapsmonge-ampereoptimalrightside

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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.

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