On the Structure of Singular Points of a Solution to Newton’s Least Resistance Problem

Abstract

We consider the following problem stated in 1993 by Buttazzo and Kawohl (Math Intell 15:7–12, 1993): minimize the functional $${\int \limits } {\int \limits }_{\Omega } (1 + |\nabla u(x,y)|^{2})^{-1} dx dy$$ in the class of concave functions u : Ω → [0,M], where $${\Omega } \subset \mathbb {R}^{2}$$ is a convex domain and M > 0. It generalizes the classical minimization problem, which was initially stated by I. Newton in 1687 in the more restricted class of radial functions. The problem is not solved until now; there is even nothing known about the structure of singular points of a solution. In this paper we, first, solve a family of auxiliary 2D least resistance problems and, second, apply the obtained results to study singular points of a solution to our original problem. More precisely, we derive a necessary condition for a point being a ridge singular point of a solution and prove, in particular, that all ridge singular points with horizontal edge lie on the top level and zero level sets.