# 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 -

## References

top- 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. Zbl0808.60008
- N. Ahmad, The geometry of shape recognition via the Monge-Kantorovich optimal transport problem. Ph.D. dissertation (2004).
- L.A. Caffarelli and V.I. Oliker, Weak solutions of one inverse problem in geometric optics. Preprint (1994). Zbl1202.78003
- 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. Zbl0917.65104
- W. Gangbo, Quelques problèmes d'analyse non convexe. Habilitation à diriger des recherches en mathématiques. Université de Metz (Janvier 1995).
- W. Gangbo and R.J. McCann, The geometry of optimal transportation. Acta Math.177 (1996) 113–161. Zbl0887.49017
- W. Gangbo and R. McCann, Shape recognition via Wasserstein distance. Quart. Appl. Math.58 (2000) 705–737. Zbl1039.49038
- 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. Zbl1049.49030
- 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. Zbl1129.49309
- Pengfei Guan and Xu-Jia Wang, On a Monge-Ampère equation arising in geometric optics J. Differential Geometry48 (1998) 205–223. Zbl0979.35052
- H.G. Kellerer, Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete67 (1984) 399–432. Zbl0535.60002
- B.E. Kinber, On two reflector antennas. Radio Eng. Electron. Phys.7 (1962) 973–979.
- M. Knott and C.S. Smith, On the optimal mapping of distributions. J. Optim. Theory Appl.43 (1984) 39–49. Zbl0519.60010
- E. Newman and V.I. Oliker, Differential-geometric methods in design of reflector antennas. Symposia Mathematica35 (1994) 205–223. Zbl0815.35115
- 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.
- 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. Zbl0645.35033
- V.I. Oliker, On the geometry of convex reflectors. PDE's, Submanifolds and Affine Differential Geometry, Banach Center Publications57 (2002) 155–169. Zbl1020.52005
- R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton (1970). Zbl0193.18401
- L. Rüschendorf, On c-optimal random variables. Appl. Stati. Probab. Lett.27 (1996) 267–270. Zbl0847.62046
- C. Smith and M. Knott, On Hoeffding-Fréchet bounds and cyclic monotone relations. J. Multivariate Anal.40 (1992) 328–334. Zbl0745.62055
- X.-J. Wang, On design of a reflector antenna. Inverse Problems12 (1996) 351–375. Zbl0858.35142
- X.-J. Wang, On design of a reflector antenna II. Calculus of Variations and PDE's20 (2004) 329–341.
- B.S. Westcott, Shaped Reflector Antenna Design. Research Studies Press, Letchworth, UK (1983).
- 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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