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

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 $C^1$ cost implies a locally Lipschitz free boundary. As an application, we address a problem discussed by Caffarelli and McCann \cite{CM} regarding cost functions satisfying the Ma-Trudinger-Wang condition (A3): if the non-negative source density is in some $L^p({\mathbb{R}}^n)$ space for $p\in (\frac{n+1}{2},\mathrm{\infty }]$ and the positive target density is bounded away from zero, then the free boundary is a semiconvex $C_{loc}^{1,\alpha }$ hypersurface. Furthermore, we show that a locally Lipschitz cost implies a rectifiable free boundary and initiate a corresponding regularity theory in the Riemannian setting.