2019/06/05 by Alexander Plakhov, Plakhov, Alexander
Mathematics · #26B25 #49Q10 #52A15 #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1906.02320
openalex publication_date 2019/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The aim of this paper is twofold. First, we cut off a part of a convex surface by a plane near a ridge point and characterize the limiting behavior of the surface measure in S2 induced by this part of surface when the plane approaches the point. Second, this characterization is applied to Newton's least resistance problem for convex bodies: minimize the functional ∫∫Ω(1 + |∇ u(x,y)|2)-1 dx dy in the class of convex functions u: Ω→ [0,M], where Ω⊂ R2 is a convex body and M > 0. It has been known that if u_* solves the problem then |∇ u_*(x,y)| ≥ 1 at all regular points (x,y) such that u_*(x,y) > 0. We prove that if the lower level set L0 = \ (x,y): u_*(x,y) = 0 \ has nonempty interior, then for almost all points of its boundary ( x, y) ∈ ∂ L0 one has lim_\stackrel(x,y)→( x, y)u_*(x,y)>0|∇ u_*(x,y)| = 1.