In this paper we develop a boundary ε-regularity theory for optimal transport maps between bounded open sets with $C^{1,\alpha }$-boundary. Our main result asserts sharp $C^{1,\alpha }$-regularity of transport maps at the boundary in form of a linear estimate under certain assumptions: The main quantitative assumptions are that the local nondimensionalized transport cost is small and that the boundaries are locally almost flat in $C^{1,\alpha }$. Our method is completely variational and builds on the recently developed interior regularity theory.

中文翻译：

---

最优运输图的尖锐边界ε-正则性

在本文中，我们为有界开放集之间的最优输运图开发了边界ε-正则性理论$C^{\mathrm{1个}，\alpha }$-边界。我们的主要结果断言$C^{\mathrm{1个}，\alpha }$-在某些假设下以线性估计的形式在边界上的运输图的规律性：主要的定量假设是，当地无量纲的运输成本很小，并且边界在当地几乎平坦 $C^{\mathrm{1个}，\alpha }$。我们的方法是完全可变的，并基于最近开发的内部规律性理论。