Hardy's inequality in a limiting case on general bounded domains

Abstract

In this paper, we study Hardy's inequality in a limiting case: integral(Omega) vertical bar del u vertical bar(N) dx >= C-N(Omega) integral(Omega) vertical bar u(x)vertical bar(N)/vertical bar x vertical bar(N)(log R/vertical bar x vertical bar)(N) dx for functions u is an element of W-0(1,N) (Omega), where Omega is a bounded domain in RN with R = sup(x is an element of Omega)vertical bar x vertical bar. We study the attainability of the best constant C-N(Omega) in several cases. We provide sufficient conditions that assure C-N(Omega) > CN(B-R) and CN(Omega) is attained, here B-R is the N-dimensional ball with center the origin and radius R. Also, we provide an example of Omega subset of R-2 such that C-2(Omega) > C-2(B-R) = 1/4 and C-2(Omega) is not attained.