$W^{2,1}$ regularity for solutions of the Monge-Amp\`ere equation

By Guido De Philippis, Alessio Figalli. 2011

83 citations recorded by the source corpus.

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 $k\in \mathbb N$.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/188fa0da547ee4c496429a38426c09f8380e2276

How researchers see this paper

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In Applied Mathematics

1. Bodhisattva Sen
2. Haigang Li
3. J. Bao
4. Lei Zhang
5. Nabarun Deb
6. Yanyan Li
7. Aobing Li

In Geometry and Topology

1. Bodhisattva Sen
2. Haigang Li
3. J. Bao
4. Lei Zhang
5. Nabarun Deb

Citations in the corpus

• Monge–Ampère equation on exterior domains (2013, background): “J. Bao (B) · H. Li Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Ministry of Education, Beijing 100875, China e-mail: jgbao@bnu.edu.cn H. Li e-mail: hgli@bnu.edu.cn L. Zhang Department of Mathematics, University of Florida, 358 Little Hall”
• Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation (2019, background): “Since such discussions are beyond the scope of this article, we would like to refer the interested reader to De Philippis and Figalli (2013), Caffarelli (1990), and Villani (2009, chap.”