On generalized Newton’s aerodynamic problem

Plakhov, A.

doi:10.1090/mosc/318

On generalized Newton’s aerodynamic problem
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by A. Plakhov

Trans. Moscow Math. Soc. 2021, 183-191

DOI: https://doi.org/10.1090/mosc/318

Published electronically: March 15, 2022

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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 (see G. Buttazzo, V. Ferone, and B. Kawohl [Math. Nachr. 173 (1995), pp. 71–89]) 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 $(\bar {x}, \bar {y}) \in \partial L$ one has $\lim _{\substack {(x,y)\to (\bar {x},\bar {y})\\u(x,y)<M}}|\nabla u(x,y)| = 1$. As a by-product, we obtain a result concerning local properties of convex surfaces near ridge points.

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