Method of nose stretching in Newton's problem of minimal resistance

Abstract

We consider the problem $\inf\big\{ \int\!\!\int_\Omega (1 + |\nabla u(x_1,x_2)|^2)^{-1} dx_1 dx_2 : \text{ the function } u : \Omega \to \mathbb{R} \text{ is concave and } 0 \le u(x) \le M \text{ for all } x = (x_1, x_2) \in \Omega =\{ |x| \le 1 \} \, \big\}$ (Newton's problem) and its generalizations. In the paper by Brock, Ferone, and Kawohl (1996) it is proved that if a solution $u$ is $C^2$ in an open set $\mathcal{U} \subset \Omega$ then $\det D^2u = 0$ in $\mathcal{U}$. It follows that graph$(u)\rfloor_\mathcal{U}$ does not contain extreme points of the subgraph of $u$.

In this paper we prove a somewhat stronger result. Namely, there exists a solution $u$ possessing the following property. If $u$ is $C^1$ in an open set $\mathcal{U} \subset \Omega$ then graph$(u\rfloor_\mathcal{U})$ does not contain extreme points of the convex body $C_u = \{ (x,z) :\, x \in \Omega,\ 0 \le z \le u(x) \}$. As a consequence, we have $C_u = \text{\rm Conv} (\overline{\text{\rm Sing$C_u$}})$, where Sing$C_u$ denotes the set of singular points of $\partial C_u$. We prove a similar result for a generalization of Newton's problem.