Skip to main content
SpringerLink
Log in

Problems of Optimal Mass Transportation

  • Chapter
  • First Online:
Exterior Billiards

  • 616 Accesses

Abstract

In this chapter, several special problems of optimal mass transportation (or Monge–Kantorovich problems) are considered. Their solution will be used in the next chapter when we solve problems of minimum and maximum resistance for nonconvex and rough bodies.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
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

Bibliography

  1. Aleksenko, A., Plakhov, A.: Bodies of zero resistance and bodies invisible in one direction. Nonlinearity22, 1247–1258 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L.: Lecture notes on optimal transport problems. In: Mathematical Aspects of Evolving Interfaces (Funchal, Portugal, 2000), ed. P. Colli and J.F. Rodrigues, Lecture Notes in Math.1812, Springer, Berlin, 2003, 1–52

    Google Scholar 

  3. Bachurin, P., Khanin, K., Marklof, J., Plakhov, A.: Perfect retroreflectors and billiard dynamics. J. Mod. Dyn.5, 33–48 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Belloni, M., Kawohl, B.: A paper of Legendre revisited. Forum Math.9, 655–668 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Belloni, M., Wagner, A.: Newton’s problem of minimal resistance in the class of bodies with prescribed volume. J. Convex Anal.10, 491–500 (2003)

    MathSciNet  MATH  Google Scholar 

  6. Borg, K.I., Söderholm, L.H.: Orbital effects of the Magnus force on a spinning spherical satellite in a rarefied atmosphere. Eur. J. Mech. B/Fluids27, 623–631 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Borg, K.I., Söderholm, L.H., Essén, H.: Force on a spinning sphere moving in a rarefied gas. Phys. Fluids15, 736–741 (2003)

    Article  Google Scholar 

  8. Brock, F., Ferone, V., Kawohl, B.: A symmetry problem in the calculus of variations. Calc. Var.4, 593–599 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations and Their Applications65, Birkhäuser, Boston, Inc., Boston, MA (2005)

    Google Scholar 

  10. Bunimovich. L.B.: Mushrooms and other billiards with divided phase space. Chaos11, 802–808 (2001)

    Google Scholar 

  11. Buttazzo, G., Ferone, V., Kawohl, B.: Minimum problems over sets of concave functions and related questions. Math. Nachr.173, 71–89 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Buttazzo, G., Kawohl, B.: On Newton’s problem of minimal resistance. Math. Intell.15, 7–12 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chechkin, G.A., Piatnitski, A.L., Shamaev, A.S.: Homogenization. Methods and Applications. Translations of Mathematical Monographs, vol. 234. American Mathematical Society, Providence (2007)

    Google Scholar 

  14. Chernov, N.: Entropy, Lyapunov exponents, and mean free path for billiards. J. Stat. Phys.88, 1–29 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chow, T.S.: Wetting of rough surfaces. J. Phys.: Condens. Matter10(27), L445 (1998)

    Google Scholar 

  16. Comte, M., Lachand-Robert, T.: Newton’s problem of the body of minimal resistance under a single-impact assumption. Calc. Var. Partial Differ. Eq.12, 173–211 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Comte, M., Lachand-Robert, T.: Existence of minimizers for Newton’s problem of the body of minimal resistance under a single-impact assumption. J. Anal. Math.83, 313–335 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Comte, M., Lachand-Robert, T.: Functions and domains having minimal resistance under a single-impact assumption. SIAM J. Math. Anal.34, 101–120 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Eaton, J.E.: On spherically symmetric lenses. Trans. IRE Antennas Propag.4, 66–71 (1952)

    Google Scholar 

  20. Fieseler, P.D.: A method for solar sailing in a low Earth orbit. Acta Astron.43, 531–541 (1998)

    Article  Google Scholar 

  21. Glutsyuk, A., Kudryashov, Yu.: On quadrilateral orbits in planar billiards. Dokl. Akad. Nauk438, 590–592 (2011) (in Russian)

    Google Scholar 

  22. Glutsyuk, A.A, Kudryashov, Yu.G: On quadrilateral orbits in planar billiards. Doklady Math.83, 371–373 (2011) [translated from Dokl. Akad. Nauk 438, 509–592 (2011) (Russian)]

    Google Scholar 

  23. Gouveia, P.D.F.: Computação de Simetrias Variacionais e Optimização da Resistência Aerodinâmica Newtoniana. Ph.D. Thesis, Universidade de Aveiro (2007)

    Google Scholar 

  24. Gouveia, P., Plakhov, A., Torres, D.: Two-dimensional body of maximum mean resistance. Appl. Math. Comput.215, 37–52 (2009). doi:10.1016/j.amc.2009.04.030

    Article  MathSciNet  MATH  Google Scholar 

  25. Gutkin, E.: Billiard dynamics: a survey with the emphasis on open problems. Regul. Chaotic Dyn.8, 1–13 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Harrison, I.K., Swinerd, G.G.: A free molecule aerodynamic investigation using multiple satellite analysis. Planet. Space Sci.44, 171–180 (1996)

    Article  Google Scholar 

  27. Ivanov, S.G., Yanshin, A.M.: Forces and moments acting on bodies rotating around a symmetry axis in a free molecular flow. Fluid Dyn.15, 449 (1980)

    Article  MATH  Google Scholar 

  28. Ivrii, V.Ya.: The second term of the spectral asymptotics for a laplace-beltrami operator on manifolds with boundary. Funct. Anal. Appl.14, 98–106 (1980)

    Google Scholar 

  29. Lachand-Robert, T., Oudet, E. Minimizing within convex bodies using a convex hull method. SIAM J. Optim.16, 368–379 (2006)

    Article  MathSciNet  Google Scholar 

  30. Lachand-Robert, T., Peletier, M.A.: Newton’s problem of the body of minimal resistance in the class of convex developable functions. Math. Nachr.226, 153–176 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lachand-Robert, T., Peletier, M.A.: An example of non-convex minimization and an application to Newton’s problem of the body of least resistance. Ann. Inst. H. Poincaré Anal. Non Lin.18, 179–198 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Legendre, A.M.: Sur la maniére de distinguer les maxima des minima dans le calcul des variations. Memoires de L’Academie royale de Sciences annee 1786, pp. 7–37. Paris (1788)

    Google Scholar 

  33. Levin, V.L.: Solution of the Monge and the Monge–Kantorovich problems. Theory and applications. Dokl. Akad. Nauk388, 7–10 (2003) (in Russian)

    Google Scholar 

  34. Levin, V.L.: Optimality conditions and exact solutions of the two-dimensional Monge–Kantorovich problem. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI),312 (2004) [Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. XI. Translation in J. Math. Sci. (N.Y.)133, 1456–1463 (2006) (in Russian)]

    Google Scholar 

  35. Levin, V.L.: Optimal solutions of the Monge problem. Adv. Math. Econ.6, 85–122 (2004)

    Article  Google Scholar 

  36. Matheron, G.: Random Sets and Integral Geometry. Wiley, New York (1975)

    MATH  Google Scholar 

  37. McCann, R.J.: Exact solutions to the transportation problem on the line. Proc. R. Soc. Lond. A455, 1341–1380 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  38. Mehta, R.D.: Aerodynamics of sport balls. Annu. Rev. Fluid Mech.17, 151–189 (1985)

    Article  Google Scholar 

  39. Moe, K., Moe, M.M.: Gas–surface interactions and satellite drag coefficients. Planet. Space Sci.53, 793–801 (2005)

    Article  Google Scholar 

  40. Newton, I.: Philosophiae naturalis principia mathematica. London: Streater (1687)

    Google Scholar 

  41. Ogilvy, J.A.: Theory of Wave Scattering from Random Rough Surfaces. A. Hilger in Bristol, England, Philadelphia (1991)

    MATH  Google Scholar 

  42. Persson, B.N.J.: Contact mechanics for randomly rough surfaces. Surf. Sci. Rep.61, 201–227 (2006)

    Article  MathSciNet  Google Scholar 

  43. Plakhov, A.: Newton’s problem of a body of minimal aerodynamic resistance. Doklady Math.390, 314–317 (2003)

    MathSciNet  Google Scholar 

  44. Plakhov, A.: Newton’s problem of the body of minimal resistance with a bounded number of collisions. Russ. Math. Surv.58, 191–192 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  45. Plakhov, A.: Newton’s problem of the body of minimum mean resistance. Sbornik: Math.195, 1017–1037 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  46. Plakhov, A.: Precise solutions of the one-dimensional Monge–Kantorovich problem. Sbornik: Math.195, 1291–1307 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  47. Plakhov, A.: Newton’s problem of minimal resistance for bodies containing a half-space. J. Dyn. Contr. Syst.10, 247–251 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  48. Plakhov, A., Torres, D.: Newton’s aerodynamic problem in media of chaotically moving particles. Sbornik: Math.196, 885–933 (2005). doi:10.1070/SM2005v196n06ABEH000904

    Article  MathSciNet  MATH  Google Scholar 

  49. Plakhov, A.: Billiards in unbounded domains reversing the direction of motion of a particle. Russ. Math. Surv.61, 179–180 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  50. Plakhov, A.: Billiards and two-dimensional problems of optimal resistance. Arch. Ration. Mech. Anal.194, 349–382 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  51. Plakhov, A.: Billiard scattering on rough sets: two-dimensional case. SIAM J. Math. Anal.40, 2155–2178 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  52. Plakhov, A.: Scattering in billiards and problems of Newtonian aerodynamics. Russ. Math. Surv.64, 873–938 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  53. Plakhov, A.: Comment on “Functions and domains having minimal resistance under a single-impact assumption” [SIAM J. Math. Anal. 34 (2002), pp.101–120]. SIAM J. Math. Anal.41, 1721–1724 (2009). doi:10.1137/09075439X

    Google Scholar 

  54. Plakhov, A.: Mathematical retroreflectors. Discr. Contin. Dyn. Syst. A30, 1211–1235 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  55. Plakhov, A.: Optimal roughening of convex bodies. Can. J. Math.64, 1058–1074 (2012)

    Article  MathSciNet  Google Scholar 

  56. Plakhov, A., Gouveia, P.: Problems of maximal mean resistance on the plane. Nonlinearity20, 2271–2287 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  57. Plakhov, A., Roshchina, V.: Invisibility in billiards. Nonlinearity24, 847–854 (2011)

    MathSciNet  MATH  Google Scholar 

  58. Plakhov, A., Tchemisova, T.: Force acting on a spinning rough disk in a flow of non-interacting particles. Doklady Math.79, 132–135 (2009)

    Article  Google Scholar 

  59. Plakhov, A., Tchemisova, T., Gouveia, P.: Spinning rough disk moving in a rarefied medium. Proc. R. Soc. A.466, 2033–2055 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  60. Plakhov, A., Torres, D.: Newton’s aerodynamic problem in media of chaotically moving particles. Sbornik: Math.196, 885–933 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  61. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley (Interscience), New York (1962)

    MATH  Google Scholar 

  62. Prandtl, L.: Application of the “Magnus Effect” to the wind propulsion of ships. Die. Naturwiss.13, 93–108 (1926) [Transl. NACA-TM-367]

    Google Scholar 

  63. Rachev, S.T., Rüschendorf, L.: Mass Transportation Problems, vol. 1: Theory. Probability and its Applications (New York). Springer, New York (1998)

    Google Scholar 

  64. Rubinov, S.I., Keller, J.B.: The transverse force on a spinning sphere moving in a viscous fluid. J. Fluid Mech.11, 447–459 (1961)

    Article  MathSciNet  Google Scholar 

  65. Rychlik, M.R.: Periodic points of the billiard ball map in a convex domain. J. Diff. Geom.30, 191–205 (1989)

    MathSciNet  MATH  Google Scholar 

  66. Santaló, L.A.: Integral Geometry and Geometric Probability. Addison-Wesley, Reading (1976)

    MATH  Google Scholar 

  67. Stojanov, L.: Note on the periodic points of the billiard. J. Diff. Geom.34, 835–837 (1991)

    MathSciNet  MATH  Google Scholar 

  68. Tabachnikov, S.: Billiards. Société Mathématique de France, Paris (1995)

    MATH  Google Scholar 

  69. Tikhomirov, V.M.: Newton’s aerodynamic problem. Kvant (5), 11–18 (1982) (in Russian)

    Google Scholar 

  70. Tikhomirov, V.M.: Stories about maxima and minima. In: Mathematical World, vol. 1. AMS, Providence (1990) [Translated from: V.M. Tikhomirov. Rasskazy o maksimumakh i minimumakh. Nauka, Moscow, 1986 (in Russian)]

    Google Scholar 

  71. Tyc, T., Leonhardt, U.: Transmutation of singularities in optical instruments. New J. Phys.10, 115038 (8 pp.) (2008)

    Google Scholar 

  72. Uckelmann, L.: Optimal couplings between one-dimensional distributions. In: Benes, V., Stepan, J. (eds.) Distributions with Given Marginals and Moment Problems, pp. 275–281. Kluwer, Dordrecht (1997)

    Chapter  Google Scholar 

  73. Vorobets, Ya.B.: On the measure of the set of periodic points of a billiard. Math. Notes55, 455–460 (1994)

    Google Scholar 

  74. Wang, C.T.: Free molecular flow over a rotating sphere. AIAA J.10, 713 (1972)

    Article  Google Scholar 

  75. Weidman, P.D., Herczynski, A.: On the inverse Magnus effect in free molecular flow. Phys. Fluids16, L9–L12 (2004)

    Article  MathSciNet  Google Scholar 

  76. Wojtkowski, M.P.: Principles for the design of billiards with nonvanishing Lyapunov exponents. Commun. Math. Phys.105, 391–414 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  77. Wojtkowski, M.P.: Two applications of Jacobi fields to the billiard ball problem. J. Diff. Geom.40, 155–164 (1994)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Aveiro, Aveiro, Portugal

    Alexander Plakhov

  2. Institute for Information Transmission Problems, Moscow, Russia

    Alexander Plakhov

Authors
  1. Alexander Plakhov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media New York

About this chapter

Cite this chapter

Plakhov, A. (2012). Problems of Optimal Mass Transportation. In: Exterior Billiards. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4481-7_5

Download citation

Publish with us

Policies and ethics

Search

Navigation