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

Let $u$ minimize the functional $F(u)={\int }_{\mathrm{\Omega }}f(\mathrm{\nabla }u(x))dx$ in the class of convex functions $u:\mathrm{\Omega }\to \mathbb{R}$ 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) , $\text{is zero}$ }$beingaclosednowheredensesetin${\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 $F(u)={\int }_{\mathrm{\Omega }}(1+\mathrm{\mid }\mathrm{\nabla }u(x){\mathrm{\mid }}^2{)}^{-1}dx$.