Existence of optimal maps in the reflector-type problems

¹ School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA; gangbo@math.gatech.edu
² Dept. of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA; oliker@mathcs.emory.edu

Received: 4 May 2005
Revised: 9 September 2005

Abstract

In this paper, we consider probability measures μ and ν on a d-dimensional sphere in and cost functions of the form that generalize those arising in geometric optics where We prove that if μ and ν vanish on -rectifiable sets, if |l'(t)|>0, and is monotone then there exists a unique optimal map T_o that transports μ onto where optimality is measured against c. 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 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.