Free boundary regularity in the optimal partial transport problem

Abstract

In the optimal partial transport problem, one is asked to transport a fraction $0<m\le \min \{||f|{|}_{L^1},||g|{|}_{L^1}\}$ 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.

32 pages, 2 figures, accepted for publication in JFA