2002/01/24 by Yanyan Li, YanYan Li, Li, YanYan +2
Computer Science · Mathematics · #35J60 #58E35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP #math.DG #msc:35J60 #msc:58E35
paper · pdf · doi:10.48550/arxiv.math/0201232
35 pages
arxiv created 2002/01/24 · openalex publication_date 2002/01/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (M,g) be a smooth compact Riemannian manifold without boundary of dimension n>=6. We prove that align* ‖u‖L2^*(M,g)2 ≤ K2∫M\|∇g u|2+c(n)Rgu2\dvg +A‖u‖L2n/(n+2)(M,g)2, align* for all u∈ H1(M), where 2^*=2n/(n-2), c(n)=(n-2)/[4(n-1)], Rg is the scalar curvature, K-1=inf‖∇ u‖L2(\Rn)‖u‖L2n/(n-2)(\Rn)-1 and A>0 is a constant depending on (M,g) only. The inequality is \em sharp in the sense that on any (M,g), K can not be replaced by any smaller number and Rg can not be replaced by any continuous function which is smaller than Rg at some point. If (M,g) is not locally conformally flat, the exponent 2n/(n+2) can not be replaced by any smaller number. If (M,g) is locally conformally flat, a stronger inequality, with 2n/(n+2) replaced by 1, holds in all dimensions n>=3.