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

De Philippis, Guido; Figalli, Alessio

doi:10.2140/apde.2013.6.993

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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

DOI: 10.2140/apde.2013.6.993

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 $W_{l o c}^{2, 1}$ if their right-hand sides converge strongly in $L_{l o c}^{1}$ . As a corollary, we deduce strong $W_{l o c}^{1, 1}$ stability of optimal transport maps.

Keywords

Monge–Ampère, stability, Sobolev convergence

Mathematical Subject Classification 2010

Primary: 35J96

Secondary: 35B45

Milestones

Received: 19 March 2012

Revised: 27 September 2012

Accepted: 15 November 2012

Published: 21 August 2013

Authors

Guido De Philippis
Scuola Normale Superiore p.za dei Cavalieri 7 I-56126 Pisa Italy

Alessio Figalli
Department of Mathematics University of Texas at Austin 1 University Station C1200 Austin, TX 78712 United States

Vol. 6, No. 4, 2013