$W^{2,1}$ regularity for solutions of the Monge-Ampère equation

De Philippis Guido - Figalli Alessio

year: 2012
journal: Invent. Math.
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 $W^{2,1}_{\rm loc}$. This is obtained by showing higher integrability a-priori estimates for $D^2u$, namely $D^2 u \in L \log^k L$ for any $k \in \mathbb N$.

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