Existence of Optimal Maps in the Reflector-type Problems

In this paper, we consider probability measures μ and ν on a d–dimensional sphere in R, d ≥ 1, and cost functions of the form c(x,y) = l( |x−y| 2 2 ) that generalize those arising in geometric optics where l(t) = − log t. We prove that if μ and ν vanish on (d− 1)–rectifiable sets, if |l′(t)| > 0, limt→0+ l(t) = +∞, and g(t) := t(2 − t)(l′(t))2 is monotone then there exists a unique optimal map To that transports μ onto ν, where optimality is measured against c. Furthermore, infx |Tox − x| > 0. Our approach is based on direct variational arguments. In the special case when l(t) = − log t, existence of optimal maps on the sphere was obtained earlier in [8] and [22] under more restrictive assumptions. In these studies, it was assumed that either μ and ν are absolutely continuous with respect to the d–dimensional Haussdorff measure, or they have disjoint supports. Another aspect of interest in this work is that it is in contrast with the work in [7] where it is proved that when l(t) = t then existence of an optimal map fails when μ and ν are supported by Jordan surfaces. School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 (gangbo@math. gatech.edu). WG gratefully acknowledges the support of National Science Foundation grants DMS00-74037, and DMS-02-00267. Dept. of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA, ([email protected]). The research of VO was partially supported by a grant from Emory University Research Committee and by the National Science Foundation grant DMS-04-05622.