Calculus of Variations and Geometric Measure Theory
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G. De Philippis - A. Figalli

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

created by dephilipp on 22 Nov 2011
modified by figalli on 14 Nov 2012


Accepted Paper

Inserted: 22 nov 2011
Last Updated: 14 nov 2012

Journal: Invent. Math.
Year: 2012


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 $k \in \mathbb N$.

Tags: GeMeThNES
Keywords: Monge-Ampère equation, Sobolev regularity, higher integrability, a-priori estimates


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