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

Abstract

We study the best possible constants c(n) in the Brezis-Marcus inequalities for u∈H01(Bn) in balls Bn={x∈Rn:|x-x0|<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 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→∞. 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,∞), 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 jν is the first zero of the Bessel function Jν of order ν and the constants 1-ν2m24andm2jν24 are sharp for all admissible values of parameters m and ν. © 2012 Elsevier Ltd.