2014 / March Volume 9 No.1
Optimal Transportation on the Hemisphere
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2014 / March
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Title | Optimal Transportation on the Hemisphere |
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Pagination | 25-44 |
Abstract | In this paper, we study the optimal transportation on the hemisphere,
with the cost function $c(x,y)=\frac12d^2(x,y)$, where $d$ is the Riemannian distance of the round sphere. The potential function satisfies a Monge-Ampere type equation with natural boundary condition. In this critical case, the hemisphere does not satisfy the c-convexity assumption. We obtain the $\it {a \ priori}$ oblique derivative estimate, and in the special case of two dimensional hemisphere, we obtain the boundary $C^2$ estimate.
Our proof does not require the smoothness of densities.
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