Optimal (partial) transport to non-convex polygonal domains

In this paper, we investigate optimal (partial) transport problems for which the target is a non-convex polygonal domain in $\mathbb{R}^2$. For the complete optimal transport problem, we prove that the singular set is locally a smooth one-dimensional curve away from finitely many points. For the optimal partial transport problem, we prove that the free boundary is smooth away from finitely many singular points. In higher dimensions, we formulate two conjectures concerning the structure of singularities when the target is a non-convex polytope.