On the best constants for the Brezis–Marcus inequalities in balls

We study the best possible constants c(n)c(n) in the Brezis–Marcus inequalities ∫Bn|∇u|2dx≥14∫Bn|u|2(ρ−|x−x0|)2dx+c(n)ρ2∫Bn|u|2dx for u∈H01(Bn) in balls Bn={x∈Rn:|x−x0|<ρ}Bn={x∈Rn:|x−x0|<ρ}. The quantity c(1)c(1) is known by our paper [F.G. Avkhadiev, K.-J. Wirths, Unified Poincaré and Hardy inequalities with sharp constants for convex domains, ZAMM Z. Angew. Math. Mech. 87 (8–9) 26 (2007) 632–642]. In the present paper we prove the estimate c(2)≥2c(2)≥2 and the assertion limn→∞c(n)n2=14, which gives that the known lower estimates in [G. Barbatis, S. Filippas, and A. Tertikas in Comm. Cont. Math. 5 (2003), no. 6, 869–881] for c(n),n≥3, are asymptotically sharp as n→∞n→∞. Also, for the 3-dimensional ball B30={x∈R3:|x|<1} we obtain a new Brezis–Marcus type inequality which contains two parameters m∈(0,∞)m∈(0,∞), ν∈(0,1/m)ν∈(0,1/m) and has the following form ∫B30|∇u(x)|2dx≥14∫B30{1−ν2m2(1−|x|)2+m2jν2(1−|x|)2−m}|u(x)|2dx, where jνjν is the first zero of the Bessel function JνJν of order νν and the constants 1−ν2m24andm2jν24 are sharp for all admissible values of parameters mm and νν.