Boundary regularity of optimal transport paths

Abstract

The optimal transport problem aims at finding an optimal way to transport a given probability measure into another. In contrast to the well-known Monge–Kantorovich problem, the ramified optimal transportation problem aims at modeling a tree-typed branching transport network by an optimal transport path between two given probability measures. An essential feature of such a transport path is to favor group transportation in a large amount. In previous works, the author has studied the existence of optimal transport paths between probability measures as well as their interior regularity, that is away from the supports of the measures. Such an optimal transport path may be understood as a 1-dimensional rectifiable current that has boundary the difference of the 2 measures and that minimizes a suitable cost functional. In this article, we study the regularity of such an optimal transport path nearby its boundary. Motivated from observing the vein structure of a tree leaf, we show that each superlevel set of an optimal transport path is locally supported on a bi-Lipschitz graph, which is a finite union of bi-Lipschitz curves.