Optimal (partial) transport to non-convex polygonal domains

Abstract: In this paper, we investigate the optimal (partial) transport problems when the target is a non-convex polygonal domain in (\mathbb{R}^2). For the complete optimal transportation, we prove that the singular set is locally a 1-dimensional smooth curve, except for a finite number of points. For the optimal partial transportation, we prove that the free boundary is smooth except for a finite number of singular points.