Optimal transportation on the hemisphere

In this paper, we study the optimal transportation on the hemisphere, with the cost function c(x, y) = 1/2d2 (x, y), where d is the Riemannian distance of the round sphere. The potential function satisfies a Monge-Ampere type equation with natural boundary condtion. In this critical case, the hemisphere does not satisfy the c-convexity assumption. We obtain the priori oblique derivate estimate, and in the special case of two demensional hemisphere, we obtain the boundary C2 estimate. Our proof does not require the smoothness of densities.