Existence of optimal maps in the reflector-type problems

Gangbo, Wilfrid; Oliker, Vladimir

ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 1, pp. 93-106.

In this paper, we consider probability measures $\mu$ and $\nu$ on a $d$-dimensional sphere in ${𝐑}^{d+1},d\ge 1,$ and cost functions of the form $c(𝐱,𝐲)=l\left(\frac{{|𝐱-𝐲|}^2}{2}\right)$ that generalize those arising in geometric optics where $l\left(t\right)=-logt.$ We prove that if $\mu$ and $\nu$ vanish on $(d-1)$-rectifiable sets, if $|l^{\text{'}}\left(t\right)|>0,$ ${lim}_{t\to 0^{+}}l\left(t\right)=+\infty ,$ and $g\left(t\right):=t(2-t){\left(l^{\text{'}}\left(t\right)\right)}^2$ is monotone then there exists a unique optimal map $T_o$ that transports $\mu$ onto $\nu ,$ where optimality is measured against $c.$ Furthermore, ${inf}_{𝐱}|T_o𝐱-𝐱|>0.$ Our approach is based on direct variational arguments. In the special case when $l\left(t\right)=-logt,$ existence of optimal maps on the sphere was obtained earlier in [Glimm and Oliker, J. Math. Sci. 117 (2003) 4096-4108] and [Wang, Calculus of Variations and PDE’s 20 (2004) 329-341] 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 the work in [Gangbo and McCann, Quart. Appl. Math. 58 (2000) 705-737] where it is proved that when $l\left(t\right)=t$ then existence of an optimal map fails when $\mu$ and $\nu$ are supported by Jordan surfaces.

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