The regularity of the free boundary in optimal transportation is equivalent to that of the potential function along the free boundary. By establishing new geometric estimates of the free boundary and studying the second boundary value problem of the Monge-Ampère equation, we obtain the $C^{2,\alpha }$ regularity of the potential function as well as that of the free boundary, thereby resolve an open problem raised by Caffarelli and McCann.

中文翻译：

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C2,α$C^{2,\alpha }$ 最优运输中自由边界的正则性

最优输运中自由边界的正则性等价于沿自由边界的势函数的正则性。通过建立新的自由边界几何估计并研究Monge-Ampère方程的第二边值问题，我们得到${\mathrm{VS}}^{2,\alpha }$势函数的正则性以及自由边界的正则性，从而解决了 Caffarelli 和 McCann 提出的开放问题。