Conditions for existence of single valued optimal transport maps on convex boundaries with nontwisted cost

We prove that if $\Omega\subset \mathbb{R}^{n+1}$ is a (not necessarily strictly) convex, $C^1$ domain, and $\mu$ and $\bar{\mu}$ are probability measures absolutely continuous with respect to surface measure on $\partial \Omega$, with densities bounded away from zero and infinity, whose $2$-Monge-Kantorovich distance is sufficiently small, then there exists a continuous Monge solution to the optimal transport problem with cost function given by the quadratic distance on the ambient space $\mathbb{R}^{n+1}$. This result is also shown to be sharp, via a counterexample when $\Omega$ is uniformly convex but not $C^1$. Additionally, if $\Omega$ is $C^{1, \alpha}$ regular for some $\alpha$, then the Monge solution is shown to be H\"older continuous.

我们证明，如果$ \ omega \ subset \ mathbb {r}^{n+1} $是（不一定是严格的）凸，$ c^1 $域，以及$ \ mu $和$ \ mu $和$ \ bar {\ mu} $是在$ partientive and yountial and fortivity and Inftientive and Inftientivity and in eftientive and Infrigity and g的范围内的范围，\\ $ 2 $ -MONGE-KANTOROVICH距离足够小，然后存在一个连续的Monge解决方案，用于最佳传输问题，其成本函数在环境空间上由二次距离给出的成本函数$ \ mathbb {r}^{n+1} $。当$ \ omega $均匀凸出而不是$ c^1 $时，通过反例显示，此结果也很清晰。此外，如果$ \ omega $是$ c^{1，\ alpha} $对于某些$ \ alpha $的常规，则Monge解决方案被证明是H \ h \“较旧的连续。