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The Refractor Problem in Reshaping Light Beams

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References

  1. Caffarelli L.A., Gutiérrez C.E., Huang Q.: On the regularity of reflector antennas. Ann. Math. 167, 299–323 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Caffarelli, L.A., Huang, Q.: Reflector problem in Rn endowed with non-Euclidean norm. Arch. Rational Mech. Anal. (to appear)

  3. Caffarelli, L.A., Oliker, V.: Weak solutions of one inverse problem in geometric optics. Preprint, 1994

  4. Descartes, R.: Discourse on Method, Optics, Geometry, and Meteorology. Hackett Publishing Co., 2001. (Translated, with an introduction, by Paul J. Olscamp)

  5. Gangbo W., Oliker W.: Existence of optimal maps in the reflector-type problems. ESAIM Control Optim. Calc. Var. 13(1), 93–106 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Guan P., Wang X.-J.: On a Monge–Ampère equation arising from optics. J. Differ. Geom. 48, 205–222 (1998)

    Article  MATH  Google Scholar 

  7. Kline, M., Kay, I.W.: Electromagnetic theory and geometrical optics. In: Pure and Applied Mathematics, vol. XII. Wiley, London, 1965

  8. Loeper, G.: On the regularity of maps solutions of optimal transportation problems, 2006. http://arxiv.org/pdf/math/0504137

  9. Ma X.-N., Trudinger N., Wang X.-J.: Regularity of potential functions of the optimal transportation problem. Arch. Rational Mech. Anal. 177(2), 151–183 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Oliker V., Glimm T.: Optical design of single reflector systems and the Monge–Kantorovich mass transfer problem. J. Math. Sci. 117, 4096–4108 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Trudinger, N.: Recent developments in elliptic partial differential equations of Monge–Ampère type. In: Proceedings of the International Congress of Mathematicians, Madrid, Spain. European Mathematical Society, pp. 291–301, 2006

  12. Trudinger, N., Wang, X.-J.: On the second boundary value problem for equations of Monge–Ampère type equations and optimal transportation, 2007. http://arxiv.org/pdf/math/0601086

  13. Villani, C.: Topics in optimal transportation. In: Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence, RI, 2003

  14. Villani, C.: Optimal transport, old and new, 2007. http://www.umpa.ens-lyon.fr/~cvillani/Cedrif/B07B.StFlour.pdf

  15. Wang X.-J.: On the design of a reflector antenna. Inverse Probl. 12, 351–375 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Wang X.-J.: On the design of a reflector antenna II. Calc. Var. Partial Differ. Equ. 20(3), 329–341 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Cristian E. Gutiérrez.

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Communicated by L.C. Evans

Cristian E. Gutiérrez was partially supported by NSF Grant DMS-0610374.

Qingbo Huang was partially supported by NSF Grant DMS-0502045.

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Gutiérrez, C.E., Huang, Q. The Refractor Problem in Reshaping Light Beams. Arch Rational Mech Anal 193, 423–443 (2009). https://doi.org/10.1007/s00205-008-0165-x

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