考虑将一个质量分布输送到另一个质量分布上，选择该输送以优化总期望成本，其中从x输送到y的每单位质量的成本由光滑函数c（x，y）给出。如果源密度f^+（x）在开区域U' \subset R^n中远离零和无穷大有界，并且目标密度f^-（y）在其支集clV \subset R^n上远离零和无穷大有界，该支集相对于U'是强c凸的，并且运输成本c是非负交叉曲线的，我们推导出U'内部最优映射的连续性和内射性（使得关联势u属于C^1（U'））。这个结果在低/内部正则性设置中提供了关键的一步：在随后的论文[15]中，我们使用它来建立关于球面的任意乘积上的黎曼距离平方的最优映射的正则性。本文还提供了Figalli和Loeper在较弱的（实际上是必要的）假设A3 w [17]下得出最优映射的二维连续性结论所需的论证。在高维中，如果密度f^ \pm是Hölder连续的，我们的结果允许从Liu，Trudinger和Wang [33]的工作中推导出映射在U'内的连续可微性（实际上，相关势的C^{2，α}_{loc}正则性）.
英文摘要：
Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x,y). If the source density f^+(x) is bounded away from zero and infinity in an open region U' \subset R^n, and the target density f^-(y) is bounded away from zero and infinity on its support clV \subset R^n, which is strongly c-convex with respect to U', and the transportation cost c is non-negatively cross-curved, we deduce continuity and injectivity of the optimal map inside U' (so that the associated potential u belongs to C^1(U')). This result provides a crucial step in the low/interior regularity setting: in a subsequent paper [15], we use it to establish regularity of optimal maps with respect to the Riemannian distance squared on arbitrary products of spheres. The present paper also provides an argument required by Figalli and Loeper to conclude in two dimensions continuity of optimal maps under the weaker (in fact, necessary) hypothesis A3w [17]. In higher dimensions, if the densities f^\pm are Hölder continuous, our result permits continuous differentiability of the map inside U' (in fact, C^{2,α}_{loc} regularity of the associated potential) to be deduced from the work of Liu, Trudinger and Wang [33].
