Free boundary regularity in the optimal partial transport problem

In the optimal partial transport problem, one is asked to transport a fraction $0<m \leq \min\{||f||_{L^1}, ||g||_{L^1}\}$ of the mass of $f=f \chi_\Omega$ onto $g=g\chi_\Lambda$ while minimizing a transportation cost. If $f$ and $g$ are bounded away from zero and infinity on strictly convex domains $\Omega$ and $\Lambda$, respectively, and if the cost is quadratic, then away from $\partial(\Omega \cap \Lambda)$ the free boundaries of the active regions are shown to be $C_{loc}^{1,\alpha}$ hypersurfaces up to a possible singular set. This improves and generalizes a result of Caffarelli and McCann \cite{CM} and solves a problem discussed by Figalli \cite[Remark 4.15]{Fi}. Moreover, a method is developed to estimate the Hausdorff dimension of the singular set: assuming $\Omega$ and $\Lambda$ to be uniformly convex domains with $C^{1,1}$ boundaries, we prove that the singular set is $\mathcal{H}^{n-2}$ $\sigma$-finite in the general case and $\mathcal{H}^{n-2}$ finite if $\Omega$ and $\Lambda$ are separated by a hyperplane.