Title: Partial regularity of optimal transport with Coulomb cost Show Chinese title

Title: 最优传输的局部正则性与库仑代价

Abstract: We prove that for two-marginal optimal transport with Coulomb cost, the optimal map is a $C^{1,\alpha}$ diffeomorphism outside a closed set of Lebesgue measure zero provided the marginals are $\alpha$-H\"older continuous and bounded away from zero and infinity. Excluding a set of measure zero is necessary as optimal maps for the Coulomb cost have long been known to exhibit jump singularities across codimension $1$ surfaces (even for smooth marginals on convex domains). Show Chinese abstract

Abstract: 我们证明，对于具有库仑成本的两边缘最优传输，如果边缘是$\alpha$-霍尔德连续且远离零和无穷大的情况下，最优映射在勒贝格测度为零的闭集之外是一个$C^{1,\alpha}$微分同胚。排除测度为零的集合是必要的，因为对于库仑成本的最优映射早已知道会在余维$1$的表面上表现出跳跃奇异性（即使对于凸域上的光滑边缘也是如此）。