Existence of optimal maps in the reflector-type problems

Abstract

In this paper, we consider probability measures $\mu$ and $\nu$ on a $d$--dimensional sphere in $\Rd, d \geq 1,$ and cost functions of the form $c(\x,\y)=l(\frac{|\x-\y|^2}{2})$ that generalize those arising in geometric optics where $l(t)=-\log t.$ We prove that if $\mu$ and $\nu$ vanish on $(d-1)$--rectifiable sets, if $|l'(t)|>0,$ $\lim_{t\to 0^+}l(t)=+\infty,$ and $g(t):=t(2-t)(l'(t))^2$ is monotone then there exists a unique optimal map $T_o$ that transports $\mu$ onto $\nu,$ where optimality is measured against $c.$ Furthermore, $\inf_{\x}|T_o\x-\x|>0.$ 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 by Glimm-Oliker and independently by X.-J. Wang under more restrictive assumptions. In these studies, it was assumed that either $\mu$ and $\nu$ 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 a result by Gangbo-McCann who proved that when $l(t)=t$ then existence of an optimal map fails when $\mu$ and $\nu$ are supported by Jordan surfaces.