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Trudy Moskovskogo Matematicheskogo Obshchestva, 2021, Volume 82, Issue 1, Pages 217–226 (Mi mmo656)  

This article is cited in 2 scientific papers (total in 2 papers)

On generalized Newton's aerodynamic problem

A. Plakhovab

a Institute for Information Transmission Problems, Moscow, Russia
b Department of Mathematics, University of Aveiro, Portugal
References:
Abstract: We consider the generalized Newton's least resistance problem for convex bodies: minimize the functional $\iint_\Omega (1 + |\nabla u(x,y)|^2)^{-1} dx dy$ in the class of concave functions $u\colon \Omega \to [0,M]$, where the domain $\Omega \subset \mathbb{R}^2$ is convex and bounded and $M > 0$. It has been known [1] that if $u$ solves the problem then $|\nabla u(x,y)| \ge 1$ at all regular points $(x,y)$ such that $u(x,y) < M$. We prove that if the upper level set $L = \{ (x,y)\colon u(x,y) = M \}$ has nonempty interior, then for almost all points of its boundary $(\overline{x}, \overline{y}) \in \partial L$ one has $\lim_{\substack{(x,y)\to(\overline{x}, \overline{y})\\\ u(x,y)<M}}|\nabla u(x,y)| = 1$. As a by-product, we obtain a result concerning local properties of convex surfaces near ridge points.
Key words and phrases: convex body, surface area measure, Newton's problem of minimal resistance.
Funding agency Grant number
Fundação para a Ciência e a Tecnologia UIDB/04106/2020
UIDP/04106/2020
This work is supported by The Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology, references UIDB/04106/2020 and UIDP/04106/2020.
Received: 28.02.2021
English version:
Transactions of the Moscow Mathematical Society, 2021, Volume 82, Pages 183–191
DOI: https://doi.org/10.1090/mosc/318
Bibliographic databases:
Document Type: Article
UDC: 517.988.38
MSC: 52A15, 26B25, 49Q10
Language: English
Citation: A. Plakhov, “On generalized Newton's aerodynamic problem”, Tr. Mosk. Mat. Obs., 82, no. 1, MCCME, M., 2021, 217–226; Trans. Moscow Math. Soc., 82 (2021), 183–191
Citation in format AMSBIB
\Bibitem{Pla21}
\by A.~Plakhov
\paper On generalized Newton's aerodynamic problem
\serial Tr. Mosk. Mat. Obs.
\yr 2021
\vol 82
\issue 1
\pages 217--226
\publ MCCME
\publaddr M.
\mathnet{http://mi.mathnet.ru/mmo656}
\transl
\jour Trans. Moscow Math. Soc.
\yr 2021
\vol 82
\pages 183--191
\crossref{https://doi.org/10.1090/mosc/318}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85124418518}
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  • https://www.mathnet.ru/eng/mmo/v82/i1/p217
  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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