2021/09/29 by Plakhov, Alexander
#49Q10 #52A15 #52A40 #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2109.14207
Let u minimize the functional F(u) = ∫Ωf(∇ u(x)) dx in the class of convex functions u : Ω→ \mathbb R satisfying 0 ≤ u ≤ M, where Ω⊂ \mathbb R2 is a compact convex domain with nonempty interior and M > 0, and f : \mathbb R2 → \mathbb R is a C2 function, with \ ξ: the smallest eigenvalue of f"(ξ) is zero \ being a closed nowhere dense set in \mathbb R2. 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) = ∫Ω(1 + |∇ u(x)|2)-1 dx.