On generalized Newton's aerodynamic problem

Abstract

We consider the generalized Newton's least resistance problem for convex bodies: minimize the functional $\int\!\!\int_\Omega (1 + |\nabla u(x,y)|^2)^{-1} dx\, dy$ in the class of concave functions $u: \Omega \to [0,M]$, where the domain $\Omega \subset \mathbb{R}^2$ is convex and bounded and $M > 0$. It has been known \cite{BFK} 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 $L = \{ (x,y): u(x,y) = M \}$ has nonempty interior, then for almost all points of its boundary $(\bar x, \bar y) \in \pl L$ one has $\lim_{\stackrel{(x,y)\to(\bar x,\bar y)}{u(x,y)<M}}|\nabla u(x,y)| = 1$. As a by-product, we obtain a result concerning local properties of convex surfaces near ridge points.