2021
DOI: 10.48550/arxiv.2109.14207
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A solution to Newton's least resistance problem is uniquely defined by its singular set

Abstract: Let u minimize the functional F (u) = Ω f (∇u(x)) dx in the class of convex functions u : Ω → R satisfying 0 ≤ u ≤ M , where Ω ⊂ R 2 is a compact convex domain with nonempty interior and M > 0, and f : R 2 → R is a C 2 function, with {ξ : det f ′′ (ξ) = 0} being a closed nowhere dense set in 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… Show more

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