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

Abstract

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In this paper, we consider probability measures μ and ν on a d-dimensional sphere in 𝐑 d + 1 , d 1 , and cost functions of the form c ( 𝐱 , 𝐲 ) = l ( | 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 0 + l ( t ) = + , and g ( t ) : = t ( 2 - t ) ( l ' ( t ) ) 2 is monotone then there exists a unique optimal map To 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 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.

How to cite

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Gangbo, 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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