2022/03/27 by Alexander Plakhov, Plakhov, Alexander
Computer Science · Engineering · Mathematics · #26B25 #49Q10 #52A15 #Advanced Mathematical Modeling in Engineering #Composite Material Mechanics #FOS: Mathematics #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2203.14235
openalex publication_date 2022/03/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the following problem stated in 1993 by Buttazzo and Kawohl: minimize the functional ∫ ∫Ω(1 + |∇ u(x,y)|2)-1 dx dy in the class of concave functions u: Ω→ [0,M], where Ω⊂ ℝ2 is a convex domain and M > 0. It generalizes the classical minimization problem, which was initially stated by I. Newton in 1687 in the more restricted class of radial functions. The problem is not solved until now; there is even nothing known about the structure of singular points of a solution. In this paper we, first, solve a family of auxiliary 2D least resistance problems and, second, apply the obtained results to study singular points of a solution to our original problem. More precisely, we derive a necessary condition for a point being a ridge singular point of a solution and prove, in particular, that all ridge singular points with horizontal edge lie on the top level and zero level sets.