On the regularity of the free boundary in the optimal partial transport problem

2013/03/11 by Shibing Chen, Emanuel Indrei, Chen, Shibing +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #math.AP #math.FA

paper · pdf · doi:10.48550/arxiv.1303.2715

Submitted

arxiv created 2013/12/11 · arxiv updated 2013/12/12

Abstract

This paper concerns the regularity and geometry of the free boundary in the optimal partial transport problem for general cost functions. More specifically, we prove that a C¹ cost implies a locally Lipschitz free boundary. As an application, we address a problem discussed by Caffarelli and McCann \citeCM regarding cost functions satisfying the Ma-Trudinger-Wang condition (A3): if the non-negative source density is in some L^p(ℝⁿ) space for p ∈ ((n+1)/(2),∞] and the positive target density is bounded away from zero, then the free boundary is a semiconvex C_loc^1,α hypersurface. Furthermore, we show that a locally Lipschitz cost implies a rectifiable free boundary and initiate a corresponding regularity theory in the Riemannian setting.

Cited by

• Unidimensional semi-discrete partial optimal transport

Related

• Regularity of free boundaries in optimal transportation
• C^2,α regularity of free boundaries in optimal transportation
• Free boundary regularity in the optimal partial transport problem
• Partial Regularity for optimal transport maps
• Dynamics of Optimal Partial Transport