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

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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 u∈Llogk L for any k∈ℕ.

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Authors and Affiliations

  1. Scuola Normale Superiore, p.za dei Cavalieri 7, 56126, Pisa, Italy

    Guido De Philippis

  2. Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX, 78712, USA

    Alessio Figalli

Authors
  1. Guido De Philippis
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  2. Alessio Figalli
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Correspondence to Alessio Figalli.

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De Philippis, G., Figalli, A. W 2,1 regularity for solutions of the Monge–Ampère equation. Invent. math. 192, 55–69 (2013). https://doi.org/10.1007/s00222-012-0405-4

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  • DOI: https://doi.org/10.1007/s00222-012-0405-4

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