2023/07/01 by Shibing Chen, Jiakun Liu, Chen, Shibing +3
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2307.00262
openalex publication_date 2023/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we establish the global W2,p estimate for the Monge-Ampère obstacle problem: (Du)\sharpfχ_\u>(1)/(2)|x|2\=g, where f and g are positive continuous functions supported in disjoint bounded C2 uniformly convex domains Ω and Ω^*, respectively. Furthermore, we assume that ∫Ωf≥ ∫Ω^*g. The main result shows that Du: U→Ω^*, where U=\u>(1)/(2)|x|2\, is a W1, p diffeomorphism for any p∈(1,∞). Previously, it was only known to be a continuous homeomorphism according to Caffarelli and McCann \citeCM. It is worth noting that our result is sharp, as we can construct examples showing that even with the additional assumption of smooth densities, the optimal map Du is not Lipschitz. This obstacle problem arises naturally in optimal partial transportation.