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

We study the best possible constants c(n) in the Brezis-Marcus inequalities integral(Bn) vertical bar del u vertical bar(2) dx >= 1/4 integral(Bn) vertical bar u vertical bar(2)/(rho - vertical bar x - x(0)vertical bar)(2) dx + c(n)/rho(2) integral(Bn) vertical bar u vertical bar(2) dx for u is an element of H-0(1)(B-n) in balls B-n = {x is an element of R-n : vertical bar x - x(0)vertical bar < rho} The quantity c(1) is known by our paper [F.G. Avkhadiev, K.-J. Wirths, Unified Poincare and Hardy inequalities with sharp constants for convex domains, ZAMM Z. Angew. Math. Mech. 87 (8-9) 26 (2007) 632-6421. In the present paper we prove the estimate c(2) >= 2 and the assertion lim(n ->infinity) 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 ->infinity . Also, for the 3-dimensional ball B-3(0) = {x is an element of R-3 : vertical bar x vertical bar < 1) we obtain a new Brezis-Marcus type inequality which contains two parameters m is an element of (0, infinity), v is an element of (0, 1/m) and has the following form integral(B30) vertical bar del u(x)vertical bar(2) dx >= 1/4 integral(B30) {1 - v(2)m(2)/(1 - vertical bar x vertical bar)(2) + m(2)j(v)(2)/(1 - vertical bar x vertical bar)(2-m)}vertical bar u(x)vertical bar(2) dx, where j(v) is the first zero of the Bessel functionl, of order v and the constants 1- v(2)m(2)/4 and m(2)j(v)(2)/4 are sharp for all admissible values of parameters m and v. (C) 2012 Elsevier Inc. All rights reserved.