In this paper, we consider probability measures and on a -dimensional sphere in and cost functions of the form that generalize those arising in geometric optics where We prove that if and vanish on -rectifiable sets, if and is monotone then there exists a unique optimal map that transports onto where optimality is measured against Furthermore, Our approach is based on direct variational arguments. In the special case when 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 and are absolutely continuous with respect to the -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 then existence of an optimal map fails when and are supported by Jordan surfaces.
Keywords: mass transport, reflector problem, Monge-Ampere equation
@article{COCV_2007__13_1_93_0, author = {Gangbo, Wilfrid and Oliker, Vladimir}, title = {Existence of optimal maps in the reflector-type problems}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {93--106}, publisher = {EDP-Sciences}, volume = {13}, number = {1}, year = {2007}, doi = {10.1051/cocv:2006017}, mrnumber = {2282103}, zbl = {1136.49015}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2006017/} }
TY - JOUR AU - Gangbo, Wilfrid AU - Oliker, Vladimir TI - Existence of optimal maps in the reflector-type problems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2007 SP - 93 EP - 106 VL - 13 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2006017/ DO - 10.1051/cocv:2006017 LA - en ID - COCV_2007__13_1_93_0 ER -
%0 Journal Article %A Gangbo, Wilfrid %A Oliker, Vladimir %T Existence of optimal maps in the reflector-type problems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2007 %P 93-106 %V 13 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2006017/ %R 10.1051/cocv:2006017 %G en %F COCV_2007__13_1_93_0
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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