Skip to main content Accessibility help

Existence of optimal maps in the reflector-type problems

  • Wilfrid Gangbo (a1) and Vladimir Oliker (a2)


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 T o 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's 20 (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.



Hide All
[1] Abdellaoui, T. and Heinich, H., Sur la distance de deux lois dans le cas vectoriel. C.R. Acad. Sci. Paris Sér. I Math. 319 (1994) 397400.
[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] Caffarelli, L.A. and Kochengin, S. and Oliker, V.I., On the numerical solution of the problem of reflector design with given far-field scattering data. Cont. Math. 226 (1999) 1332.
[5] W. Gangbo, Quelques problèmes d'analyse non convexe. Habilitation à diriger des recherches en mathématiques. Université de Metz (Janvier 1995).
[6] Gangbo, W. and McCann, R.J., The geometry of optimal transportation. Acta Math. 177 (1996) 113161.
[7] Gangbo, W. and McCann, R., Shape recognition via Wasserstein distance. Quart. Appl. Math. 58 (2000) 705737.
[8] Glimm, T. and Oliker, V.I., Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem. J. Math. Sci. 117 (2003) 40964108.
[9] Glimm, T. and Oliker, V.I., Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat's principle. Indiana Univ. Math. J. 53 (2004) 12551278.
[10] Pengfei Guan and Xu-Jia Wang, On a Monge-Ampère equation arising in geometric optics J. Differential Geometry 48 (1998) 205–223.
[11] Kellerer, H.G., Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete 67 (1984) 399432.
[12] Kinber, B.E., On two reflector antennas. Radio Eng. Electron. Phys. 7 (1962) 973979.
[13] Knott, M. and Smith, C.S., On the optimal mapping of distributions. J. Optim. Theory Appl. 43 (1984) 3949.
[14] Newman, E. and Oliker, V.I., Differential-geometric methods in design of reflector antennas. Symposia Mathematica 35 (1994) 205223.
[15] Norris, A.P. and Westcott, B.S., Computation of reflector surfaces for bivariate beamshaping in the elliptic case. J. Phys. A: Math. Gen 9 (1976) 21592169.
[16] Oliker, V.I. and Waltman, P., 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) 361374.
[17] V.I. Oliker, On the geometry of convex reflectors. PDE's, Submanifolds and Affine Differential Geometry, Banach Center Publications 57 (2002) 155–169.
[18] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton (1970).
[19] Rüschendorf, L., On c-optimal random variables. Appl. Stati. Probab. Lett. 27 (1996) 267270.
[20] Smith, C. and Knott, M., Hoeffding-Fréchet, On bounds and cyclic monotone relations. J. Multivariate Anal. 40 (1992) 328334.
[21] Wang, X.-J., On design of a reflector antenna. Inverse Problems 12 (1996) 351375.
[22] Wang, X.-J., On design of a reflector antenna II. Calculus of Variations and PDE's 20 (2004) 329341.
[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.


Existence of optimal maps in the reflector-type problems

  • Wilfrid Gangbo (a1) and Vladimir Oliker (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.