Inventiones mathematicae

, Volume 192, Issue 1, pp 55-69

First online:

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

  • Guido De PhilippisAffiliated withScuola Normale Superiore
  • , Alessio FigalliAffiliated withDepartment of Mathematics, The University of Texas at Austin Email author 

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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.