Additional singularities in minimizers of Newton’s minimal resistance problem

Determine whether minimizers u of the Cartesian Newton minimal resistance problem min{∫Ω [1/(1+|∇u(x)|^2)] dx : u ∈ C_M}, with C_M = {u: Ω → ℝ concave, 0 ≤ u ≤ M}, exhibit singularities (discontinuities of ∇u) beyond the boundary of the plateau {x ∈ Ω : u(x) = M}; and, if such singularities exist, characterize their precise structure and location within Ω.

Background

In the Cartesian formulation, admissible shapes are described by concave functions u: Ω → ℝ bounded by 0 ≤ u ≤ M, and resistance is given by ∫Ω 1/(1+|∇u|^2) dx. A theorem in Section 2 shows that at differentiable points of a minimizer u, either ∇u(x) = 0 or |∇u(x)| ≥ 1, implying no globally regular (C^1) solutions.

The remark notes that along the upper boundary of the plateau {u = M}, the gradient satisfies |∇u| ≥ 1, forcing a region of singularity (discontinuous ∇u). It then explicitly states an open question about whether there are additional singularities elsewhere in the domain and, if so, what their structure and location are.