Inventiones mathematicae

, Volume 192, Issue 1, pp 55–69

W2,1 regularity for solutions of the Monge–Ampère equation

Authors

  • Guido De Philippis
    • Scuola Normale Superiore
    • Department of MathematicsThe University of Texas at Austin
Article

DOI: 10.1007/s00222-012-0405-4

Cite this article as:
De Philippis, G. & Figalli, A. Invent. math. (2013) 192: 55. doi:10.1007/s00222-012-0405-4

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 D2u, namely D2uLlogkL for any k∈ℕ.

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© Springer-Verlag 2012