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Weak solutions of one inverse problem in geometric optics. (English. Russian original) Zbl 1202.78003

J. Math. Sci., New York 154, No. 1, 39-49 (2008); translation from Probl. Mat. Anal. 37, 37-46 (2008).
Summary: We consider the problem of recovering a closed convex reflecting surface such that for a given point source of light (inside the convex body bounded by the surface) the reflected directions cover a unit sphere with prespecified in advance density. In analytic formulation, the problem leads to an equation of Monge-Ampère type on a unit sphere. We formulate the problem in terms of certain associated measures and establish the existence of weak solutions. Bibliography: 11 titles.

MSC:

78A05 Geometric optics
49N45 Inverse problems in optimal control
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[1] L. A. Caffarelli and V. I. Oliker, Weak Solutions of One Inverse Problem in Geometric Optics. Preprint, 1994. · Zbl 1202.78003
[2] A. D. Aleksandrov, ”On the theory of mixed volumes, I: Extension of certain concepts in the theory of convex bodies” [in Russian], Mat. Sb. 2(44) (1937), no. 5, 947–972. · Zbl 0017.42603
[3] A. D. Aleksandrov, ”On the surface function of a convex body” [in Russian], Mat. Sb. 6(48) (1939), no. 1, 167–174.
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[7] V. I. Oliker, ”Near radially symmetric solutions of an inverse problem in geometric optics,” Inverse Problems 3 (1987), 743–756. · Zbl 0658.35094 · doi:10.1088/0266-5611/3/4/017
[8] V. Oliker and P. Waltman, ”Radially symmetric solutions of a Monge-Ampère equation arising in a reflector mapping problem,” in: Lect. Notes Math. 1285, Springer-Verlag, 1986, pp. 361–374. · Zbl 0645.35033 · doi:10.1007/BFb0080616
[9] A. V. Pogorelov, The Minkowski Multidimensional Problem, Wiley, New York, 1978. · Zbl 0387.53023
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[11] S. T. Yau, ”Open problems in geometry,” in: Proc. Symposia in Pure Mathematics, R. Greene and S. T. Yau, Eds. 54, Part 1, Am. Math. Soc., 1993, pp. 1–28. · Zbl 0801.53001
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