Calculus of Variations and Geometric Measure Theory

G. De Philippis - A. Figalli

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

created by dephilipp on 22 Nov 2011
modified on 30 Oct 2017


Accepted Paper

Inserted: 22 nov 2011
Last Updated: 30 oct 2017

Journal: Invent. Math.
Year: 2012

ArXiv: 1111.7207 PDF


In this paper we prove that a strictly convex Alexandrov solution u of the Monge-Amp\`ere equation, with right hand side bounded away from zero and infinity, is $W_{\rm loc}^{2,1}$. 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 $k\in \mathbb N$.

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