Regularity of optimal maps on the sphere: the quadratic cost and the reflector antenna

Abstract

Building on the results of Ma, Trudinger and Wang \cite{MTW}, and of the author \cite{L5}, we study two problems of optimal transportation on the sphere: the first corresponds to the cost function $d^2(x,y)$, where $d(\cdot ,\cdot )$ is the Riemannian distance of the round sphere; the second corresponds to the cost function $-\log |x-y|$, it is known as the reflector antenna problem. We show that in both cases, the {\em cost-sectional curvature} is uniformly positive, and establish the geometrical properties so that the results of \cite{L5} and \cite{MTW} can apply: global smooth solutions exist for arbitrary smooth positive data and optimal maps are Hölder continuous under weak assumptions on the data.

arXiv admin note: text overlap with arXiv:math/0504137

References in corpus (1)

• Continuity, curvature, and the general covariance of optimal transportation