PDE Seminar Abstracts

# Optimal transportation on the hemisphere

Jiakun Liu
University of Wollongong
Mon 18th Apr 2016, 2-3pm, Carslaw Room 829 (AGR)

## 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 $c\left(x,y\right)={d}^{2}\left(x,y\right)∕2$, where $d$ 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 $c$-convexity assumption. We obtain the a priori oblique derivative estimate, and in the special case of dimension two, we obtain the boundary ${C}^{2}$ estimate. Our proof does not require the smoothness of densities.

This is joint work with S.-Y. Alice Chang and Paul Yang.