Existence of optimal maps in the reflector-type problems

Gangbo, Wilfrid; Oliker, Vladimir

ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 1, pp. 93-106.

In this paper, we consider probability measures $\mu$ and $\nu$ on a $d$-dimensional sphere in ${𝐑}^{d+1},d\ge 1,$ and cost functions of the form $c(𝐱,𝐲)=l\left(\frac{{|𝐱-𝐲|}^2}{2}\right)$ that generalize those arising in geometric optics where $l\left(t\right)=-logt.$ We prove that if $\mu$ and $\nu$ vanish on $(d-1)$-rectifiable sets, if $|l^{\text{'}}\left(t\right)|>0,$ ${lim}_{t\to 0^{+}}l\left(t\right)=+\infty ,$ and $g\left(t\right):=t(2-t){\left(l^{\text{'}}\left(t\right)\right)}^2$ is monotone then there exists a unique optimal map $T_o$ that transports $\mu$ onto $\nu ,$ where optimality is measured against $c.$ Furthermore, ${inf}_{𝐱}|T_o𝐱-𝐱|>0.$ Our approach is based on direct variational arguments. In the special case when $l\left(t\right)=-logt,$ 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 $\mu$ and $\nu$ 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 $l\left(t\right)=t$ then existence of an optimal map fails when $\mu$ and $\nu$ are supported by Jordan surfaces.

Classification : 49, 35J65
Mots clés : mass transport, reflector problem, Monge-Ampere equation

[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. | Zbl

[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). | Zbl

[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. | Zbl

[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. | Zbl

[7] W. Gangbo and R. Mccann, Shape recognition via Wasserstein distance. Quart. Appl. Math. 58 (2000) 705-737. | Zbl

[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. | Zbl

[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. | Zbl

[10] Pengfei Guan and Xu-Jia Wang, On a Monge-Ampère equation arising in geometric optics J. Differential Geometry 48 (1998) 205-223. | Zbl

[11] H.G. Kellerer, Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete 67 (1984) 399-432. | Zbl

[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. | Zbl

[14] E. Newman and V.I. Oliker, Differential-geometric methods in design of reflector antennas. Symposia Mathematica 35 (1994) 205-223. | Zbl

[15] A.P. Norris and B.S. Westcott, Computation of reflector surfaces for bivariate beamshaping in the elliptic case. J. Phys. A: Math. Gen 9 (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. | Zbl

[17] V.I. Oliker, On the geometry of convex reflectors. PDE's, Submanifolds and Affine Differential Geometry, Banach Center Publications 57 (2002) 155-169. | EuDML | Zbl

[18] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton (1970). | MR | Zbl

[19] L. Rüschendorf, On $c$-optimal random variables. Appl. Stati. Probab. Lett. 27 (1996) 267-270. | Zbl

[20] C. Smith and M. Knott, On Hoeffding-Fréchet bounds and cyclic monotone relations. J. Multivariate Anal. 40 (1992) 328-334. | Zbl

[21] X.-J. Wang, On design of a reflector antenna. Inverse Problems 12 (1996) 351-375. | Zbl

[22] X.-J. Wang, On design of a reflector antenna II. Calculus of Variations and PDE's 20 (2004) 329-341. | Zbl

[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. | Zbl