On the structure of singular points of a solution to Newton's least resistance problem

TL;DR

This paper investigates the structure of singular points in solutions to a generalized Newton's least resistance problem, providing new necessary conditions for singular points and characterizing their geometric placement.

Contribution

It solves auxiliary 2D least resistance problems and applies these results to analyze the structure of singular points in the original problem.

Findings

Necessary condition for ridge singular points

All ridge singular points with horizontal edge lie on top or zero level sets

Provides insight into the geometric structure of solutions

Abstract

We consider the following problem stated in 1993 by Buttazzo and Kawohl: 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 $\Omega \subset \mathbb{R}^2$ 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.