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

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 $W_{\mathrm{loc}}^{2,1}$. This is obtained by showing higher integrability a-priori estimates for $D^{2}u$, namely $D^{2}u\in L{\log }^{k}L$ for any $k\in \mathbb{N}$.

12 pages, no figures

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