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

2011/11/30 by De Philippis, Guido, Figalli, Alessio · 3 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1111.7207

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_{\rm loc}^2,1. This is obtained by showing higher integrability a-priori estimates for D² u, namely D² u ∈ Llog^k L for any k∈ \mathbb N.

Cited by

• Partial regularity of optimal transport with Coulomb cost
• On the singular set of the free boundary for a Monge-Ampère obstacle problem
• Sharp Hausdorff Bounds for the Interior Singular Set of Convex k-Hessian Solutions

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