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 -

References

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  1. T. Abdellaoui and H. Heinich, Sur la distance de deux lois dans le cas vectoriel. C.R. Acad. Sci. Paris Sér. I Math.319 (1994) 397–400.  
  2. N. Ahmad, The geometry of shape recognition via the Monge-Kantorovich optimal transport problem. Ph.D. dissertation (2004).  
  3. L.A. Caffarelli and V.I. Oliker, Weak solutions of one inverse problem in geometric optics. Preprint (1994).  
  4. L.A. Caffarelli and S. Kochengin and V.I. Oliker, On the numerical solution of the problem of reflector design with given far-field scattering data. Cont. Math.226 (1999) 13–32.  
  5. W. Gangbo, Quelques problèmes d'analyse non convexe. Habilitation à diriger des recherches en mathématiques. Université de Metz (Janvier 1995).  
  6. W. Gangbo and R.J. McCann, The geometry of optimal transportation. Acta Math.177 (1996) 113–161.  
  7. W. Gangbo and R. McCann, Shape recognition via Wasserstein distance. Quart. Appl. Math.58 (2000) 705–737.  
  8. T. Glimm and V.I. Oliker, Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem. J. Math. Sci.117 (2003) 4096–4108.  
  9. T. Glimm and V.I. Oliker, Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat's principle. Indiana Univ. Math. J.53 (2004) 1255–1278.  
  10. Pengfei Guan and Xu-Jia Wang, On a Monge-Ampère equation arising in geometric optics J. Differential Geometry48 (1998) 205–223.  
  11. H.G. Kellerer, Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete67 (1984) 399–432.  
  12. B.E. Kinber, On two reflector antennas. Radio Eng. Electron. Phys.7 (1962) 973–979.  
  13. M. Knott and C.S. Smith, On the optimal mapping of distributions. J. Optim. Theory Appl.43 (1984) 39–49.  
  14. E. Newman and V.I. Oliker, Differential-geometric methods in design of reflector antennas. Symposia Mathematica35 (1994) 205–223.  
  15. A.P. Norris and B.S. Westcott, Computation of reflector surfaces for bivariate beamshaping in the elliptic case. J. Phys. A: Math. Gen9 (1976) 2159–2169.  
  16. V.I. Oliker and P. Waltman, Radially symmetric solutions of a Monge-Ampere equation arising in a reflector mapping problem. Proc. UAB Int. Conf. on Diff. Equations and Math. Physics, edited by I. Knowles and Y. Saito, Springer. Lect. Notes Math.1285 (1987) 361–374.  
  17. V.I. Oliker, On the geometry of convex reflectors. PDE's, Submanifolds and Affine Differential Geometry, Banach Center Publications57 (2002) 155–169.  
  18. R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton (1970).  
  19. L. Rüschendorf, On c-optimal random variables. Appl. Stati. Probab. Lett.27 (1996) 267–270.  
  20. C. Smith and M. Knott, On Hoeffding-Fréchet bounds and cyclic monotone relations. J. Multivariate Anal.40 (1992) 328–334.  
  21. X.-J. Wang, On design of a reflector antenna. Inverse Problems12 (1996) 351–375.  
  22. X.-J. Wang, On design of a reflector antenna II. Calculus of Variations and PDE's20 (2004) 329–341.  
  23. B.S. Westcott, Shaped Reflector Antenna Design. Research Studies Press, Letchworth, UK (1983).  
  24. S.T. Yau, Open problems in geometry, in Differential Geometry. Part 1: Partial Differential Equations on Manifolds (Los Angeles, 1990), R. Greene and S.T. Yau Eds., Proc. Sympos. Pure Math., Amer. Math. Soc.54 (1993) 1–28.  

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