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: minimize the functional $\int {\int }_{\Omega }(1+|\nabla u(x,y){|}^2{)}^{-1}dxdy$ 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.

arXiv comment14 pages, 6 figures