On the best constant for Hardy’s inequality in $\mathbb {R}^n$

Abstract

Let Ω be a domain in R n and p ∈ (1, ∞). We consider the (gen-eralized) Hardy inequality Ω ||u| p ≥ K Ω |u/δ| p , where δ(x) = dist (x, ∂Ω). The inequality is valid for a large family of domains, including all bounded domains with Lipschitz boundary. We here explore the connection between the value of the Hardy constant µp(Ω) = inf • W 1,p (Ω) Ω ||u| p / Ω |u/δ| p and the existence of a minimizer for this Rayleigh quotient. It is shown that for all smooth n-dimensional domains, µp(Ω) ≤ cp, where cp = (1 − 1 p) p is the one-dimensional Hardy constant. Moreover it is shown that µp(Ω) = cp for all those domains not possessing a minimizer for the above Rayleigh quotient. Finally , for p = 2, it is proved that µ 2 (Ω)