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

Abstract

We consider the problem (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 then det D 2 u = 0 in . It follows that graph 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 then graph does not contain extreme points of the convex body C u = {(x, z): x ∈ Ω, 0 ⩽ z ⩽ u(x)}. As a consequence, we have , where SingC u denotes the set of singular points of ∂C u . We prove a similar result for a generalization of Newton’s problem.