Global $W^{2,1+\epsilon }$ regularity for potentials of optimal transport of non-convex planar domains

In this paper, we investigate the optimal transport problem when the source is a non-convex polygonal domain in ${\mathbb{R}}^2$. We show a global $W^{2,1+\epsilon }$ estimate for potentials of optimal transport. Our method applies to a more general class of domains.