Existence of optimal maps in the reflector-type problems
| dc.creator | Gangbo, Wilfrid | |
| dc.creator | Oliker, Vladimir | |
| dc.date | 2007-02-25 | |
| dc.date.accessioned | 2026-07-07T07:48:48Z | |
| dc.date.available | 2026-07-07T07:48:48Z | |
| dc.description | In this paper, we consider probability measures $μ$ and $ν$ 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 $μ$ and $ν$ 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 $μ$ 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 $l(t)=-\log t,$ 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 $μ$ 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 a result by Gangbo-McCann who proved that when $l(t)=t$ then existence of an optimal map fails when $μ$ and $ν$ are supported by Jordan surfaces. | |
| dc.identifier | https://arxiv.org/abs/math/0702747 | |
| dc.identifier | http://arxiv.org/abs/math/0702747 | |
| dc.identifier | ESAIM: Control, Optimization and Calculus of Variations, Vol. 13, No. 1, 2007, pp.93-106 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/124629 | |
| dc.subject | Optimization and Control | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 49J20, 35J65 | |
| dc.title | Existence of optimal maps in the reflector-type problems | |
| dc.type | text |