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 FigalliEmail author


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.


Differential Inclusion High Integrability Doubling Measure Minkowski Problem Bounded Convex Domain 
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© 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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