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

Abstract

Let u minimize the functional $F(u) = \int _\Omega f(\nabla u(x))\, dx$ in the class of convex functions $u : \Omega \rightarrow {{\mathbb {R}}}$ satisfying $0 \le u \le M$, where $\Omega \subset {{\mathbb {R}}}^2$ is a compact convex domain with nonempty interior and $M > 0$, and $f : {{\mathbb {R}}}^2 \rightarrow {{\mathbb {R}}}$ 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 $F(u) = \int _\Omega (1 + |\nabla u(x)|^2)^{-1}\, dx$.