本课题拟深入研究次黎曼流形上的Kantorovich位势函数适定性，运用几何分析、测度论、偏微分方程理论、最优化理论的思想和方法，重点研究次黎曼流形上Monge-Ampère方程和Monge-Ampère型方程与方程组解的正则性问题；然后研究奇异测地线对位势函数适定性的影响，进而研究奇异测地线对最优映射存在唯一性的影响。近年来最优运输理论在数学学科中的变分法、偏微分方程和几何测度论等方向的研究得到了迅速发展，并在流体力学、统计物理、经济学、交通运输、图像处理、气象学和宇宙学等领域有重要应用，已越来越引起人们的广泛关注和重视。全面深入地研究次黎曼流形上Monge问题解的适定性及与之相关的数学问题对丰富最优运输问题、完全非线性偏微分方程理论以及次黎曼流形上的几何分析都有着重要意义。

We study the well-posedness of Kantorovich potentials on sub-Riemannian manifolds. These main methods and ideas used to study these problems come from geometric analysis, measure theory, partial differential equations and optimization theory. We focus on studying the regularity of solutions of Monge-Ampère and Monge-Ampère type equations and systems on sub-Riemannian manifolds. In addition, we study the relationship between singular geodesics and well-posedness of Kantorovich potentials. We also pay attention to effects of singular geodesics on existence and uniqueness of optimal transport maps. Nowadays, optimal mass transportation has developed rapidly in many mathematics branches such as calculus of variations, partial differential equations, geometric measure theory, etc. Furthermore, optimal transportation problems have wide applications in many fields such as fluid mechanics, statistical physics, economics, transportation, image processing, meteorology, cosmology and so on. Therefore, it has been raised great interests and attention of optimal mass transportation. Further research on well-posedness of solutions to the Monge problem and those related mathematics problems on sub-Riemannian manifolds may combine optimal mass transportation, partial differential equations and geometric analysis on sub-Riemannian manifolds.