On generalized Newton's aerodynamic problem

A. Plakhov

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 82 (2021) no. 1, pp. 217-226

Résumé

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 $\lim_{\substack{(x,y)\to(\overline{x}, \overline{y})\\\ u(x,y)$. As a by-product, we obtain a result concerning local properties of convex surfaces near ridge points.