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Journal of Mathematical Sciences

, Volume 154, Issue 1, pp 39–49 | Cite as

Weak solutions of one inverse problem in geometric optics

  • L. A. CaffarelliEmail author
  • V. I. Oliker
Article

Abstract

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.

Keywords

Weak Solution Unit Sphere Convex Body Radial Function Geometric Optic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    L. A. Caffarelli and V. I. Oliker, Weak Solutions of One Inverse Problem in Geometric Optics. Preprint, 1994.Google Scholar
  2. 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.Google Scholar
  3. 3.
    A. D. Aleksandrov, “On the surface function of a convex body” [in Russian], Mat. Sb. 6(48) (1939), no. 1, 167–174.zbMATHGoogle Scholar
  4. 4.
    H. Busemann, Convex Surfaces, Wiley, New York, 1958.zbMATHGoogle Scholar
  5. 5.
    B. Dacorogna and J. Moser, “On a partial differential equation involving the Jacobian determinant,” Ann. Inst. Henri Poincaré, Analyse non Linéaire 7 (1990), no. 1, 1–26.zbMATHMathSciNetGoogle Scholar
  6. 6.
    V. Galindo-Israel, W. A. Imbriale, and R. Mittra, “On the theory of the synthesis of single and dual offset shaped reflector antennas,” IEEE Trans. Antenna and Propagation 35 (1987), no. 8, 887–755.CrossRefGoogle Scholar
  7. 7.
    V. I. Oliker, “Near radially symmetric solutions of an inverse problem in geometric optics,” Inverse Problems 3 (1987), 743–756.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 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.CrossRefMathSciNetGoogle Scholar
  9. 9.
    A. V. Pogorelov, The Minkowski Multidimensional Problem, Wiley, New York, 1978.zbMATHGoogle Scholar
  10. 10.
    B. S. Westcott and A. P. Norris, “Reflector synthesis for generalized far fields,” J. Phys. A: Math. Gen. 3 (1975), no. 4, 521–532.CrossRefGoogle Scholar
  11. 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.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.University of Texas at AustinAustinUSA
  2. 2.Emory UniversityAtlantaUSA

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