Regularity of the homogeneous Monge-Ampère equation

Qi Rui Li; Xu Jia Wang

doi:10.3934/dcds.2015.35.6069

Regularity of the homogeneous Monge-Ampère equation

Qi Rui Li, Xu Jia Wang

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

Abstract

In this paper, we study the regularity of convex solutions to the Dirichlet problem of the homogeneous Monge-Ampère equation det D²u = 0. We prove that if the domain is a strip region and the boundary functions are locally uniformly convex and C^k+2,α smooth, then the solution is C^k+2,αsmooth up to boundary. By an example, we show the solution may fail to be C² smooth if boundary functions are not locally uniformly convex. Similar results have also been obtained for the Dirichlet problem on bounded convex domains.

Original language	English
Pages (from-to)	6069-6084
Number of pages	16
Journal	Discrete and Continuous Dynamical Systems
Volume	35
Issue number	12
DOIs	https://doi.org/10.3934/dcds.2015.35.6069
Publication status	Published - 1 Dec 2015

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AB - In this paper, we study the regularity of convex solutions to the Dirichlet problem of the homogeneous Monge-Ampère equation det D2u = 0. We prove that if the domain is a strip region and the boundary functions are locally uniformly convex and Ck+2,α smooth, then the solution is Ck+2,αsmooth up to boundary. By an example, we show the solution may fail to be C2 smooth if boundary functions are not locally uniformly convex. Similar results have also been obtained for the Dirichlet problem on bounded convex domains.

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Regularity of the homogeneous Monge-Ampère equation

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