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

Let $u$ minimize the functional $F(u) = \int_Ωf(\nabla u(x))\, dx$ in the class of convex functions $u : Ω\to {\mathbb R}$ satisfying $0 \le u \le M$, where $Ω\subset {\mathbb R}^2$ 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 $\{ ξ: \, \text{the smallest eigenvalue of} \, f"(ξ) \, \text{is zero} \}$ 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 $F(u) = \int_Ω(1 + |\nabla u(x)|^2)^{-1}\, dx$.

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