Existence of optimal maps in the reflector-type problems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 1, pp. 93-106.

In this paper, we consider probability measures μ and ν on a d-dimensional sphere in 𝐑 d+1 ,d1, and cost functions of the form c(𝐱,𝐲)=l(|𝐱-𝐲| 2 2) that generalize those arising in geometric optics where l(t)=-logt. We prove that if μ and ν vanish on (d-1)-rectifiable sets, if |l ' (t)|>0, lim t0 + l(t)=+, and g(t):=t(2-t)(l ' (t)) 2 is monotone then there exists a unique optimal map T o that transports μ onto ν, 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)=-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 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 [Gangbo and McCann, Quart. Appl. Math. 58 (2000) 705-737] 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.

DOI: 10.1051/cocv:2006017
Classification: 49, 35J65
Keywords: mass transport, reflector problem, Monge-Ampere equation
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Gangbo, Wilfrid; Oliker, Vladimir. Existence of optimal maps in the reflector-type problems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 1, pp. 93-106. doi : 10.1051/cocv:2006017. http://www.numdam.org/articles/10.1051/cocv:2006017/

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