Abstract
In this talk, we first give a brief introduction to the optimal transportation
problem. Then we study the optimal transportation on the hemisphere with the cost
function ,
where is
the Riemannian distance of the round sphere. The potential function
satisfies a Monge-Ampere type equation with a natural boundary
condition. In this critical case, the hemisphere does not satisfy the
-convexity
assumption. We obtain the a priori oblique derivative estimate,
and in the special case of dimension two, we obtain the boundary
estimate. Our proof does not require the smoothness of densities.
This is joint work with S.-Y. Alice Chang and Paul Yang.