Díaz Díaz, Jesús Ildefonso and Comte, M. (2005) On the Newton partially flat minimal resistance body type problems. Journal of the European Mathematical Society, 7 (4). pp. 395-411. ISSN 1435-9855
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Abstract
We study the flat region of stationary points of the functional integral(Omega) F(|del u(x)|) dx under the constraint u <= M, where Omega is a bounded domain in R-2. Here F( s) is a function which is concave for s small and convex for s large, and M > 0 is a given constant. The problem generalizes the classical minimal resistance body problems considered by Newton. We construct a family of partially flat radial solutions to the associated stationary problem when Omega is a ball. We also analyze some other qualitative properties. Moreover, we show the uniqueness of a radial solution minimizing the above mentioned functional. Finally, we consider nonsymmetric domains Omega and provide sufficient conditions which ensure that a stationary solution has a flat part.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | Newton problem; obstacle problem; quasilinear elliptic operators; flat solutions |
| Subjects: | Sciences > Mathematics > Differential geometry Sciences > Mathematics > Functional analysis and Operator theory |
| ID Code: | 15456 |
| References: | Armanini, E.: Sulla superficie di minima resistenza. Ann. Mat. Pura Appl. (3) 4, 131–149 (1900) JFM 31.0709.02 Botteron, B., Marcellini, P.: A general approach to the existence of minimizers of one-dimensional non-coercive integrals of the calculus of variations. Ann. Inst. H. Poincaré 8, 197–223 (1991) Zbl 0729.49002 MR 1096604 Brezis, H.: Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam (1973) Zbl 0252.47055 MR 0348562 Brezis, H., Kinderlehrer, D.: The smoothness of solutions to nonlinear variational inequalities. Indiana Univ. Math. J. 23, 831–844 (1974) Zbl 0278.49011 MR 0361436 Buttazzo, G., Ferone, V., Kawohl, B.: Minimum problems over sets of concave functions and related questions. Math. Nachr. 173, 71–89 (1993) Zbl 0835.49001 MR 1336954 Buttazzo, G., Kawohl, B.: On Newton's problem of minimal resistance. Math. Intelligencer 15, 7–12 (1993) Zbl 0800.49038 MR 1240664 Comte, M., Díaz, J. I.: Paper in preparation Comte, M., Lachand-Robert, T.: Newton's problem of the body of minimal resistance under a single-impact assumption. Calc. Var. 12, 173–211 (2001) Zbl 0998.49012 MR 1821236 Eggers Jr., A. J., Resnikoff, M. M., Dennis, D. H.: Bodies of revolution having minimum drag at high supersonic airspeeds. NASA Report 1306 (1958) Euler, L.: Principes généraux du mouvement des fluides. Mémoires de l'Académie des Sciences de Berlin 11, 217–273 (1755) Goldstine, H. H.: A History of the Calculus of Variations from the 17th through the 19th Century. Springer, Heidelberg (1980) Zbl 0452.49002 MR 0601774 Guasoni, P.: Problemi di ottimizzazione di forma su classi di insiemi convessi. Laurea Thesis, Univ. of Pisa (1996) Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1990) Zbl 0457.35001 MR 1786735 Lachand-Robert, T., Oudet, É.: Minimizing within convex bodies using a convex hull method. SIAM J. Optim., to appear Miele, A.: Theory of Optimum Aerodynamic Shapes. Academic Press, London (1965) Zbl 0265.76076 Newton, I.: Philosophiae Naturalis Principia Mathematica. (1686) Wagner, A.: A remark on Newton's resistance formula. Z. Angew. Math. Mech. ZAMM 79, 423–427 (1999) Zbl 0926.76101 MR 1690760 |
| Deposited On: | 01 Jun 2012 12:50 |
| Last Modified: | 01 Jun 2012 12:50 |
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