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

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 Wloc2,1 if their right-hand sides converge strongly in Lloc1. As a corollary, we deduce strong Wloc1,1 stability of optimal transport maps.

Monge–Ampère, stability, Sobolev convergence
Mathematical Subject Classification 2010
Primary: 35J96
Secondary: 35B45
Received: 19 March 2012
Revised: 27 September 2012
Accepted: 15 November 2012
Published: 21 August 2013
Guido De Philippis
Scuola Normale Superiore
p.za dei Cavalieri 7
I-56126 Pisa
Alessio Figalli
Department of Mathematics
University of Texas at Austin
1 University Station C1200
Austin, TX 78712
United States