# Existence of optimal maps in the reflector-type problems

Wilfrid Gangbo; Vladimir Oliker

ESAIM: Control, Optimisation and Calculus of Variations (2007)

- Volume: 13, Issue: 1, page 93-106
- ISSN: 1292-8119

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topGangbo, Wilfrid, and Oliker, Vladimir. "Existence of optimal maps in the reflector-type problems." ESAIM: Control, Optimisation and Calculus of Variations 13.1 (2007): 93-106. <http://eudml.org/doc/249926>.

@article{Gangbo2007,

abstract = {
In this paper, we
consider probability measures μ and ν on a d-dimensional
sphere in $\{\bf R\}^\{d+1\}, d \geq 1,$ and cost functions of the form
$c(\{\bf x\},\{\bf y\})=l(\frac\{|\{\bf x\}-\{\bf y\}|^2\}\{2\})$
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,$\lim_\{t\rightarrow 0^+\}l(t)=+\infty,$ and $g(t):=t(2-t)(l'(t))^2$ is monotone then there
exists a unique optimal map To that transports μ onto $\nu,$ where
optimality is measured against c. Furthermore, $\inf_\{\{\bf x\}\}|T_o\{\bf x\}-\{\bf x\}|>0.$ 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 in [Glimm and Oliker, J. Math. Sci.117 (2003) 4096-4108]
and [Wang, Calculus of Variations and PDE's20 (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.
},

author = {Gangbo, Wilfrid, Oliker, Vladimir},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Mass transport; reflector problem; Monge-Ampere equation.; mass transport; Monge-Ampère equation},

language = {eng},

month = {2},

number = {1},

pages = {93-106},

publisher = {EDP Sciences},

title = {Existence of optimal maps in the reflector-type problems},

url = {http://eudml.org/doc/249926},

volume = {13},

year = {2007},

}

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

DA - 2007/2//

PB - EDP Sciences

VL - 13

IS - 1

SP - 93

EP - 106

AB -
In this paper, we
consider probability measures μ and ν on a d-dimensional
sphere in ${\bf R}^{d+1}, d \geq 1,$ and cost functions of the form
$c({\bf x},{\bf y})=l(\frac{|{\bf x}-{\bf y}|^2}{2})$
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,$\lim_{t\rightarrow 0^+}l(t)=+\infty,$ and $g(t):=t(2-t)(l'(t))^2$ is monotone then there
exists a unique optimal map To that transports μ onto $\nu,$ where
optimality is measured against c. Furthermore, $\inf_{{\bf x}}|T_o{\bf x}-{\bf x}|>0.$ 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 in [Glimm and Oliker, J. Math. Sci.117 (2003) 4096-4108]
and [Wang, Calculus of Variations and PDE's20 (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.

LA - eng

KW - Mass transport; reflector problem; Monge-Ampere equation.; mass transport; Monge-Ampère equation

UR - http://eudml.org/doc/249926

ER -

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