Global regularity for the Monge-Ampère equation with natural boundary condition

2018/02/21 by Shibing Chen, Jiakun Liu, Chen, Shibing +4 · 4 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1802.07518

The exposition is polished and references are updated

openalex publication_date 2018/02/21 · arxiv created 2021/05/26 · arxiv updated 2021/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we establish the global C^2,α and W^2,p regularity for the Monge-Ampère equation det D²u = f subject to boundary condition Du(Ω) = Ω^*, where Ω and Ω^* are bounded convex domains in the Euclidean space ℝⁿ with C^1,1 boundaries, and f is a Hölder continuous function. This boundary value problem arises naturally in optimal transportation and many other applications.

Citations

• Free boundaries in optimal transport and Monge-Ampère obstacle problems
• Boundary regularity of maps with convex potentials
• Polar factorization and monotone rearrangement of vector‐valued functions
• Boundary regularity for the second boundary-value problem of Monge-Ampère equations in dimension two
• Mode Collapse and Regularity of Optimal Transportation Maps

Cited by

• Geometric invariants and the Monge-Ampere equation in Kähler geometry
• Homogeneous optimal transport maps between oblique cones
• Boundary regularity of optimal transport maps on convex domains
• Second boundary value problem for the Hessian curvature flow
• C^2,α regularity of free boundaries in optimal transportation

Related

• Global C^1,α regularity for Monge-Ampère equations on planar convex domains
• Global C^1,β and W^{2, p} regularity for some singular Monge-Ampère equations
• Boundary regularity for the second boundary-value problem of Monge-Ampère equations in dimension two
• Remarks on sharp boundary estimates for singular and degenerate Monge-Ampère equations
• On global W^2,δ estimates for the Monge-Ampère equation on general bounded convex domains