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

We study the best possible constants c ( n ) in the Brezis–Marcus inequalities ∫ B n | ∇ u | 2 d x ≥ 1 4 ∫ B n | u | 2 ( ρ − | x − x 0 | ) 2 d x + c ( n ) ρ 2 ∫ B n | u | 2 d x for u ∈ H 0 1 ( B n ) in balls B n = { x ∈ R n : | x − x 0 | < ρ } . The quantity 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 ) ≥ 2 and the assertion lim n → ∞ c ( n ) n 2 = 1 4 , 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 → ∞ . Also, for the 3-dimensional ball B 3 0 = { x ∈ R 3 : | x | < 1 } we obtain a new Brezis–Marcus type inequality which contains two parameters m ∈ ( 0 , ∞ ) , ν ∈ ( 0 , 1 / m ) and has the following form ∫ B 3 0 | ∇ u ( x ) | 2 d x ≥ 1 4 ∫ B 3 0 { 1 − ν 2 m 2 ( 1 − | x | ) 2 + m 2 j ν 2 ( 1 − | x | ) 2 − m } | u ( x ) | 2 d x , where j ν is the first zero of the Bessel function J ν of order ν and the constants 1 − ν 2 m 2 4 and m 2 j ν 2 4 are sharp for all admissible values of parameters m and ν .