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Inventiones mathematicae

, Volume 192, Issue 1, pp 55–69 | Cite as

W 2,1 regularity for solutions of the Monge–Ampère equation

  • Guido De Philippis
  • Alessio Figalli
Article

Abstract

In this paper we prove that a strictly convex Alexandrov solution u of the Monge–Ampère equation, with right-hand side bounded away from zero and infinity, is \(W^{2,1}_{\mathrm{loc}}\). This is obtained by showing higher integrability a priori estimates for D 2 u, namely D 2 uLlog k L for any k∈ℕ.

Keywords

Differential Inclusion High Integrability Doubling Measure Minkowski Problem Bounded Convex Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Scuola Normale SuperiorePisaItaly
  2. 2.Department of MathematicsThe University of Texas at AustinAustinUSA

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