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

الأصل · EN

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 CM regarding cost functions satisfying the Ma-Trudinger-Wang condition (A3): if the non-negative source density is in some Lᵖ(Rⁿ) space for p ∈ (n+1/2,∞] and the positive target density is bounded away from zero, then the free boundary is a semiconvex Cloc¹,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.