Optimal transportation on the hemisphere. (English) Zbl 1292.35129
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.
MSC:
35J60 | Nonlinear elliptic equations |
35B45 | A priori estimates in context of PDEs |
49Q20 | Variational problems in a geometric measure-theoretic setting |
28C99 | Set functions and measures on spaces with additional structure |
58J05 | Elliptic equations on manifolds, general theory |