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

Abstract

Abstract We consider the problem inf ∫ ∫ Ω ( 1 + | ∇ u ( x 1 , x 2 ) | 2 ) − 1 d x 1 d x 2 : the function u : Ω → R is concave and 0 ⩽ u ( x ) ⩽ M for all x = ( x 1 , x 2 ) ∈ Ω = { | x | ⩽ 1 } (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 U ⊂ Ω then  det  D 2 u = 0 in U . It follows that graph ( u ) 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 U ⊂ Ω then graph ( u U ) does not contain extreme points of the convex body C u = {(x, z): x ∈ Ω, 0 ⩽ z ⩽ u(x)}. As a consequence, we have C u = C o n v ( Sing C u ̄ ) , where SingC u denotes the set of singular points of ∂C u . We prove a similar result for a generalization of Newton’s problem.