A solution to Newton's least resistance problem is uniquely defined by its singular set

TL;DR

This paper proves that the optimal convex function minimizing a certain functional is uniquely determined by its singular set, with applications to Newton's classical least resistance problem.

Contribution

It establishes that the extremal points of the epigraph of the minimizer are contained in the closure of its singular points, leading to uniqueness based on the singular set.

Findings

Optimal solution is uniquely determined by its singular set.

Extremal points of the epigraph are contained in the closure of singular points.

Results apply to Newton's classical least resistance problem.

Abstract

Let $u$ minimize the functional $F(u) = \int_\Omega f(\nabla u(x))\, dx$ in the class of convex functions $u : \Omega \to {\mathbb R}$ satisfying $0 \le u \le M$, where $\Omega \subset {\mathbb R}^2$ is a compact convex domain with nonempty interior and $M > 0$, and $f : {\mathbb R}^2 \to {\mathbb R}$ is a $C^2$ function, with $\{ \xi : \, \text{the smallest eigenvalue of} \, f"(\xi) \, \text{is zero} \}$ being a closed nowhere dense set in ${\mathbb R}^2$. Let epi$(u)$ denote the epigraph of $u$. Then any extremal point $(x, u(x))$ of epi$(u)$ is contained in the closure of the set of singular points of epi$(u)$. As a consequence, an optimal function $u$ is uniquely defined by the set of singular points of epi$(u)$. This result is applicable to the classical Newton's problem, where $F(u) = \int_\Omega (1 + |\nabla u(x)|^2)^{-1}\, dx$.