Skip to main content
Log in

Existence and Regularity of the Reflector Surfaces in \({\mathbb{R}^{n+1}}\)

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

In this paper we study the problem of constructing reflector surfaces from the near field data. The light is transmitted as a collinear beam and the reflected rays illuminate a given domain on the fixed receiver surface. We consider two types of weak solutions and prove their equivalence under some convexity assumptions on the target domain. The regularity of weak solutions is a very delicate problem and the positive answer depends on a number of conditions characterizing the geometric positioning of the reflector and receiver. In fact, we show that there is a domain \({\mathcal{D}}\) in the ambient space such that the weak solution is smooth if and only if its graph lies in \({\mathcal{D}}\).

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

Access this article

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

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Abbreviations

C, C 0, C n , . . .:

Generic constants,

\({\overline{\mathcal{U}}}\) :

Closure of a set \({\mathcal{U}}\),

\({\partial \mathcal{U}}\) :

Boundary of a set \({\mathcal{U}}\) ,

\({\widehat {X}}\) :

\({\widehat X=(x^1,\ldots,x^{n},0)}\) projection of \({X=(x^1,\ldots ,x^{n+1}) \in \mathbb{R}^{n+1}}\) ,

⟨·, ·⟩:

Inner product in \({\mathbb{R}^{n+1}}\) ,

B r (x):

\({\{y \in \mathbb{R}^n: |y - x| < \}}\) ,  Open ball centered at y,

B r :

B r (0),

Γ u :

Graph of function u,

i u, D i u, Du :

\({\partial_iu=D_iu=\frac{\partial{u}}{\partial{x^i}}}\) and Du = (D 1 u, . . . , D n u),

ψ :

Defining function of receiver \({\Sigma=\{Z \in \mathbb{R}^{n+1} : \psi(Z)=0\}}\) ,

∇ψ :

(n + 1)-Dimensional gradient of receiver \({\psi: \mathbb{R}^{n+1} \longrightarrow \mathbb{R}}\),

\({\widehat{\nabla}\psi}\) :

(ψ 1, . . . ψ n ,0), Projection of ∇ψ,

Π :

Hyperplane \({\{X \in \mathbb{R}^{n+1} : x^{n+1}=0\}}\) ,

\({\mathbb{S}^{n+1}}\) :

Units sphere in \({\mathbb{R}^{n+1}}\) ,

|E|:

n-Dimensional Lebesgue measure of EΠ,

\({\mathcal{H}_{\Sigma}^{n}}\) :

n-Dimensional Hausdorff measure restricted on Σ

\({\mathbb{P}_L(\mathcal{U, V})}\) :

See (5.5),

\({\mathbb{W}^+(\mathcal{U}, \mathcal{V}), \mathbb{W}_0^+(\mathcal{U}, \mathcal{V})}\) :

See Definitions 5.1 and 5.2,

\({\mathcal{AS}^+(\mathcal{U},\Sigma)}\) :

See Definition 11.1

References

  1. Aleksandrov A.D.: Die innere Geometrie der konvexen Flächen. Akademie, Berlin (1955)

    MATH  Google Scholar 

  2. Ash R.B.: Measure, Integration and Functional Analysis. Academic Press, New York (1972)

    MATH  Google Scholar 

  3. Bakelman I.: Geometric methods for the solution of elliptic equations. Monograph, Nauka (1965)

    Google Scholar 

  4. Brenier Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  5. Caffarelli L.A.: The regularity of mappings with a convex potential. J. Am. Math. Soc. 5(1), 99–104 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Caffarelli L., Nirenberg L., Spruck J.: The Dirichlet problem for nonlinear second order elliptic equations, I: Monge–Ampère equations. Commun. Pure Appl. Math. 37, 369–402 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cheng S.Y., Yau S.T.: On the regularity of the Monge–Ampère equation det( 2 u/∂ x i ∂x j ) =  F(x, u). Comm. Pure Appl. Math. 30(1), 41–68 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  8. Karakhanyan, A.L.: The analysis and design of reflector antennae, preprint (2010)

  9. Karakhanyan A.L., Wang X.-J.: On the reflector shape design. J. Differ. Geometry 84(3), 561–610 (2010)

    MATH  MathSciNet  Google Scholar 

  10. Karakhanyan, A.L., Wang, X.-J.: The reflector design problem. Proceedings of Inter. Congress of Chinese Mathematicians 2, 1–4, 1–24 (2007)

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Liu J.: Light reflection is nonlinear optimization. Calc. Var. Partial Differ. Equ. 46(3–4), 861–878 (2013)

    Article  MATH  Google Scholar 

  13. Loeper G.: On the regularity of solutions of optimal transportation problems. Acta Mathematica 202(2), 241–283 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ma X.N., Trudinger N., Wang X.-J.: regularity of potential functions of the optimap transportation problem. Arch. Rational Mech. Anal. 177, 151–183 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. Pogorelov A.V.: Monge–Ampère equations of elliptic type. Noordhoff, Stockholm (1964)

    MATH  Google Scholar 

  16. Pogorelov A.V.: The Minkowski multidimensional problem. Wiley, New York (1978)

    MATH  Google Scholar 

  17. Pogorelov A.V.: Extrinsic geometry of convex surfaces. Translations of Mathematical Monographs. Vol. 35. AMS, Providence (1973)

  18. Rauch J., Taylor B.A.: The Dirichlet problem for the multidimensional Monge–Ampère equation. Rocky Mt. J. Math. 7(2), 345–364 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  19. Trudinger, N.S.: Lectures on nonlinear elliptic equations of second order. Lectures in Mathematical Sciences. The University of Tokyo,Tokyo (1995)

  20. Urbas, J.: Mass transfer problems. Lecture Notes. Univ. of Bonn, Bonn (1998)

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

    Article  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Aram L. Karakhanyan.

Additional information

Communicated by F. Lin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Karakhanyan, A.L. Existence and Regularity of the Reflector Surfaces in \({\mathbb{R}^{n+1}}\) . Arch Rational Mech Anal 213, 833–885 (2014). https://doi.org/10.1007/s00205-014-0743-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-014-0743-z

Keywords

Navigation