2023/08/13 by Seonghyeon Jeong, Jeong, Seonghyeon, Jun Kitagawa +1 · 1 citation
Mathematics · #35J96 #49Q22 #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.2308.06826
openalex publication_date 2023/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that if Ω⊂ ℝn+1 is a (not necessarily strictly) convex, C1 domain, and μ and μ are probability measures absolutely continuous with respect to surface measure on ∂ Ω, with densities bounded away from zero and infinity, whose 2-Monge-Kantorovich distance is sufficiently small, then there exists a continuous Monge solution to the optimal transport problem with cost function given by the quadratic distance on the ambient space ℝn+1. This result is also shown to be sharp, via a counterexample when Ω is uniformly convex but not C1. Additionally, if Ω is C1, α regular for some α, then the Monge solution is shown to be Hölder continuous.