Summary: In this paper, we study the optimal transportation on the hemisphere, with the cost function \(c(x, y)=\frac{1}{2}d^2 (x,y)\), where \(d\) is the Riemannian distance of the round sphere. The potential function satisfies a Monge-Ampère type equation with natural boundary condition. In this critical case, the hemisphere does not satisfy the c-convexity assumption. We obtain the a priori oblique estimate, and in the special case of two dimensional hemisphere, we obtain the boundary \(C^2\) estimate. Our proof does not depend on the smoothness of densities.