, Volume 192, Issue 1, pp 55-69

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

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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.