#### Vol. 6, No. 4, 2013

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

### Guido De Philippis and Alessio Figalli

Vol. 6 (2013), No. 4, 993–1000
##### 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.

##### Keywords
Monge–Ampère, stability, Sobolev convergence
Primary: 35J96
Secondary: 35B45