A survey on the Newton problem of optimal profiles

  • Giuseppe Buttazzo
Conference paper
Part of the Springer Optimization and Its Applications book series (SOIA, volume 33)


This chapter aims to present a survey on some recent results about one of the first problems in the calculus of variations, namely Newton’s problem of minimal resistance. Many variants of the problem can be studied, in relation to the various admissible classes of domains under consideration and to the different constraints that can be imposed. Here we limit ourselves essentially to the convex case. Other presentations in the workshop will deal with other kinds of domains.


Convex Body Concave Function Compactness Result Hypersonic Flow Height Class 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Belloni, A. Wagner: Newton’s problem of minimal resistance in the class of bodies with prescribed volume. J. Convex Anal., 10 (2) (2003), 491–500.MATHMathSciNetGoogle Scholar
  2. 2.
    F. Brock, V. Ferone, B. Kawohl: A symmetry problem in the calculus of variations. Calc. Var., 4 (6) (1996), 593–599.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    G. Buttazzo: Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations. Pitman Res. Notes Math. Ser. 207, Longman, Harlow (1989).Google Scholar
  4. 4.
    G. Buttazzo, V. Ferone, B. Kawohl: Minimum problems over sets of concave functions and related questions. Math. Nachr., 173 (1995), 71–89.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    G. Buttazzo, M. Giaquinta, S. Hildebrandt: One-dimensional Calculus of Variations: an Introduction. Oxford University Press, Oxford (1998).Google Scholar
  6. 6.
    G. Buttazzo, P. Gusoni: Shape optimization problems over classes of convex domains. J. Convex Anal., 4 (1997), 343–351.MATHMathSciNetGoogle Scholar
  7. 7.
    G. Buttazzo, B. Kawohl: On Newton’s problem of minimal resistance. Math. Intell., 15 (1993), 7–12.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    G. Carlier, T. Lachand-Robert: Regularity of solutions for some variational problems subject to a convexity constraint. Comm. Pure Appl. Math., 54 (5) (2001), 583–594.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    M. Comte, T. Lachand-Robert: Existence of minimizers for Newton’s problem of the body of minimal resistance under a single impact assumption. J. Anal. Math., 83 (2001), 313–335.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    M. Comte, T. Lachand-Robert: Newton’s problem of the body of minimal resistance under a single-impact assumption. Calc. Var. Partial Dif. Equations, 12 (2) (2001), 173–211.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    P. Funk: Variationsrechnung und ihre Anwendungen in Physik und Technik. Grundlehren 94, Springer-Verlag, Heidelberg (1962).Google Scholar
  12. 12.
    H. H. Goldstine: A History of the Calculus of Variations from the 17th through the 19th Century. Springer-Verlag, Heidelberg (1980).MATHGoogle Scholar
  13. 13.
    P. Guasoni: Problemi di ottimizzazione di forma su classi di insiemi convessi. Tesi di Laurea, Università di Pisa, 1995-1996.Google Scholar
  14. 14.
    W. D. Hayes, R. F. Probstein: Hypersonic Flow Theory. Academic Press, New York (1966).MATHGoogle Scholar
  15. 15.
    D. Horstmann, B. Kawohl, P. Villaggio: Newton’s aerodynamic problem in the presence of friction. NoDEA Nonlinear Diff. Equations Appl., 9 (3) (2002), 295–307.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    A. Kneser: Ein Beitrag zur Frage nach der zweckmäβigsten Gestalt der Geschoβspitzen. Archiv der Mathematik und Physik, 2 (1902), 267–278.Google Scholar
  17. 17.
    T. Lachand-Robert, E. Oudet: Minimizing within convex bodies using a convex hull method. SIAM J. Optimiz, 16 (2) (2005), 368–379.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    T. Lachand-Robert, M. A. Peletier: 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 Linéaire, 18 (2) (2001), 179–198.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    T. Lachand-Robert, M. A. Peletier: Newton’s problem of the body of minimal resistance in the class of convex developable functions. Math. Nachr., 226 (2001), 153–176.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    P. Marcellini: Nonconvex integrals of the calculus of variations. In “Methods of Nonconvex Analysis” (Varenna, 1989), Lecture Notes in Math. 1446, Springer-Verlag, Berlin (1990), 16–57.Google Scholar
  21. 21.
    A. Miele: Theory of Optimum Aerodynamic Shapes. Academic Press, New York (1965).MATHGoogle Scholar
  22. 22.
    A. Yu. Plakhov: Newton’s problem of the body of least resistance with a bounded number of collisions. Russian Math. Surveys, 58 (1) (2003), 191–192.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    A. Yu. Plakhov: Newton’s problem of minimal resistance for bodies containing a half-space. J. Dynam. Control Systems, 10 (2) (2004), 247–251.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    A. Yu. Plakhov: Newton’s problem of the body of minimal mean resistance. Sb. Math., 195 (7-8) (2004), 1017–1037.MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    A. Yu. Plakhov, D. F. M. Torres: Newton’s aerodynamic problem in media of chaotically moving particles. Sb. Math., 196 (5–6) (2005), 885–933.MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    L. Tonelli: Fondamenti di Calcolo delle Variazioni. Zanichelli, Bologna (1923).Google Scholar
  27. 27.
    N. Van Goethem: Variational problems on classes of convex domains. Commun. Appl. Anal., 8 (3) (2004), 353–371.MATHMathSciNetGoogle Scholar
  28. 28.
    A. Wagner: A remark on Newton’s resistance formula. ZAMM Z. Angew. Math. Mech., 79 (6) (1999), 423–427.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.Dipartimento di Matematica, “L. Tonelli”Università di PisaPisaItaly

Personalised recommendations