Skip to main content

The Reflector Problem and the Inverse Square Law

  • Conference paper
Geometric Methods in PDE’s

Part of the book series: Springer INdAM Series ((SINDAMS,volume 13))

  • 1217 Accesses

Abstract

We introduce a model to design reflectors that take into account the inverse square law for radiation. We prove existence of solutions in the near field case when the input and output energies are prescribed.

Al diletto amico Ermanno Lanconelli in occasione del suo settantesimo compleanno

AMS Subject Classification: 78A05, 35J96, 35Q60

Dedicated to Ermanno Lanconelli on the occasion of his 70th birthday.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    The units for this quantity are Watts because the units for \(\varOmega \subseteq S^{2}\) are considered non dimensional units, i.e., \(\varOmega\) is measured in steradians.

  2. 2.

    From [11, Theorem (11.5)] every Borel subset of X is Carathéodory measurable.

  3. 3.

    For a, b vectors in \(\mathbb{R}^{3}\), ab is the matrix a t b.

References

  1. Ambrosio, L., Tilli, P.: Topics on Analysis in Metric Spaces. Oxford Lecture Series in Mathematics and Its Applications, vol. 25. Oxford University Press, Oxford (2004)

    Google Scholar 

  2. Born, M., Wolf, E.: Principles of Optics, Electromagnetic Theory, Propagation, Interference and Diffraction of Light, 7th (expanded), 2006 edn. Cambridge University Press, Cambridge (1959)

    Google Scholar 

  3. Caffarelli, L.A., Oliker, V.: Weak solutions of one inverse problem in geometric optics. J. Math. Sci. 154(1), 37–46 (2008)

    Article  MathSciNet  Google Scholar 

  4. Gutiérrez, C.E., Huang, Q.: The near field refractor. Ann. Inst. Henri Poincaré (C) Anal. Non Linéaire. 31(4), 655–684 (2014) https://www.math.temple.edu/~gutierre/papers/nearfield.final.version.pdf

  5. Gutiérrez, C.E., Mawi, H.: The far field refractor with loss of energy. Nonlinear Anal. Theory Methods Appl. 82, 12–46 (2013)

    Article  MATH  Google Scholar 

  6. Gutiérrez, C.E., Sabra, A.: The reflector problem and the inverse square law. Nonlinear Anal. Theory Methods Appl. 96, 109–133 (2014)

    Article  MATH  Google Scholar 

  7. Kochengin, S., Oliker, V.: Determination of reflector surfaces from near-field scattering data. Inverse Prob. 13, 363–373 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. McCluney, W.R.: Introduction to Radiometry and Photometry. Artech House, Boston (1994)

    Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  11. Wheeden, R.L., Zygmund, A.: Measure and Integral. Marcel Dekker, New York (1977)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Cristian E. Gutiérrez .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Gutiérrez, C.E., Sabra, A. (2015). The Reflector Problem and the Inverse Square Law. In: Citti, G., Manfredini, M., Morbidelli, D., Polidoro, S., Uguzzoni, F. (eds) Geometric Methods in PDE’s. Springer INdAM Series, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-02666-4_15

Download citation

Publish with us

Policies and ethics