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Труды Московского математического общества, 2021, том 82, выпуск 1, страницы 217–226
(Mi mmo656)
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Эта публикация цитируется в 2 научных статьях (всего в 2 статьях)
On generalized Newton's aerodynamic problem
A. Plakhovab a Institute for Information Transmission Problems, Moscow, Russia
b Department of Mathematics, University of Aveiro, Portugal
Аннотация:
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.
Ключевые слова и фразы:
convex body, surface area measure, Newton's problem of minimal resistance.
Поступила в редакцию: 28.02.2021
Образец цитирования:
A. Plakhov, “On generalized Newton's aerodynamic problem”, Тр. ММО, 82, no. 1, МЦНМО, М., 2021, 217–226; Trans. Moscow Math. Soc., 82 (2021), 183–191
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/mmo656 https://www.mathnet.ru/rus/mmo/v82/i1/p217
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