Free boundary regularity in the optimal partial transport problem

In the optimal partial transport problem, one is asked to transport a fraction $0<m\le \min \{\mathrm{\mid }\mathrm{\mid }f\mathrm{\mid }{\mathrm{\mid }}_{L^1},\mathrm{\mid }\mathrm{\mid }g\mathrm{\mid }{\mathrm{\mid }}_{L^1}\}$ of the mass of $f=f \chi_\Omega$onto$g=g\chi_\Lambda$whileminimizingatransportationcost\mathrm{.}If$f$and$g$areboundedawayfromzeroandinfinityonstrictlyconvexdomains$\Omega$and$\Lambda$, respectively, and if the cost is quadratic, then away from $\mathrm{\partial }(\mathrm{\Omega }\cap \mathrm{\Lambda })$ 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 $\mathrm{\Omega }$ and $\mathrm{\Lambda }$ 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 $\mathrm{\Omega }$ and $\mathrm{\Lambda }$ are separated by a hyperplane.