The problem of the body of revolution of minimal resistance
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 1, pp. 206-220.

Newton's problem of the body of minimal aerodynamic resistance is traditionally stated in the class of convex axially symmetric bodies with fixed length and width. We state and solve the minimal resistance problem in the wider class of axially symmetric but generally nonconvex bodies. The infimum in this problem is not attained. We construct a sequence of bodies minimizing the resistance. This sequence approximates a convex body with smooth front surface, while the surface of approximating bodies becomes more and more complicated. The shape of the resulting convex body and the value of minimal resistance are compared with the corresponding results for Newton's problem and for the problem in the intermediate class of axisymmetric bodies satisfying the single impact assumption [Comte and Lachand-Robert, J. Anal. Math. 83 (2001) 313-335]. In particular, the minimal resistance in our class is smaller than in Newton's problem; the ratio goes to 1/2 as (length)/(width of the body) → 0, and to 1/4 as (length)/(width) → +∞.

DOI: 10.1051/cocv:2008070
Classification: 49K30, 49Q10
Keywords: newton's problem, bodies of minimal resistance, calculus of variations, billiards
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Plakhov, Alexander; Aleksenko, Alena. The problem of the body of revolution of minimal resistance. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 1, pp. 206-220. doi : 10.1051/cocv:2008070. http://archive.numdam.org/articles/10.1051/cocv:2008070/

[1] F. Brock, V. Ferone and B. Kawohl, A symmetry problem in the calculus of variations. Calc. Var. 4 (1996) 593-599. | Zbl

[2] G. Buttazzo and P. Guasoni, Shape optimization problems over classes of convex domains. J. Convex Anal. 4 (1997) 343-351. | Zbl

[3] G. Buttazzo and B. Kawohl, On Newton's problem of minimal resistance. Math. Intell. 15 (1993) 7-12. | Zbl

[4] G. Buttazzo, V. Ferone and B. Kawohl, Minimum problems over sets of concave functions and related questions. Math. Nachr. 173 (1995) 71-89. | Zbl

[5] M. Comte and T. Lachand-Robert, Newton's problem of the body of minimal resistance under a single-impact assumption. Calc. Var. 12 (2001) 173-211. | Zbl

[6] M. Comte and 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. | Zbl

[7] T. Lachand-Robert and E. Oudet, Minimizing within convex bodies using a convex hull method. SIAM J. Optim. 16 (2006) 368-379. | Zbl

[8] T. Lachand-Robert and 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. | Zbl

[9] T. Lachand-Robert and 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 (2001) 179-198. | Numdam | Zbl

[10] I. Newton, Philosophiae naturalis principia mathematica (1686). | Zbl

[11] A.Yu. Plakhov, Newton's problem of a body of minimal aerodynamic resistance. Dokl. Akad. Nauk 390 (2003) 314-317.

[12] A.Yu. Plakhov, Newton's problem of the body of minimal resistance with a bounded number of collisions. Russ. Math. Surv. 58 (2003) 191-192. | Zbl

[13] A. Plakhov and D. Torres, Newton's aerodynamic problem in media of chaotically moving particles. Sbornik: Math. 196 (2005) 885-933. | Zbl

[14] V.M. Tikhomirov, Newton's aerodynamical problem. Kvant 5 (1982) 11-18 [in Russian].

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