On the best constant for Hardy’s inequality in Rⁿ

Let Ω be a domain in Rⁿ and p∈ (1,∞ ). We consider the (generalized) Hardy inequality ∫ _Ω |∇ u|ᵖ≥ K∫ _Ω |u/δ |ᵖ, where δ (x)=distx∂ 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 μ ₚ(Ω )=inf _W1, 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, μ ₚ(Ω )≤ cₚ, where cₚ=(1-1 p)ᵖ is the one-dimensional Hardy constant. Moreover it is shown that μ ₚ(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 )<c₂=1/4 if and only if the Rayleigh quotient possesses a minimizer. Examples show that strict inequality may occur even for bounded smooth domains, but μ ₚ=cₚ for convex domains.