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

Abstract

Abstract. Let Ω be a domain in R n and p ∈ (1, ∞). We consider the (generalized) 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 constantΩ |∇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) < c 2 = 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 µp = cp for convex domains.

Геометрическое Описание Областей, Для Которых Константа Харди Равна 1/4

“…Этот факт доказан в нескольких статьях, опубликованных в 1995-1998 годах (см. [2]- [5] и обзорную статью [6]). В [2] показано, что c 2 (Ω) 1/4 для произвольной области Ω ⊂ R d , граница которой содержит хотя бы одну точку, регулярную в определенном смысле.…”

“…А именно, Е. Б. Дэвисом [2] установлено, что c 2 (Ω β ) = 1/4 для секторов вида Ω β := {re iθ ∈ C : 0 < r < 1, 0 < θ < β} при ограничении β β 0 ≈ 4.856. В [5] и [9] показано, что при любом d 3 константа Харди c 2 (Ω R1,R2 ) равна 1/4 для любого шарового слоя…”

Weighted Sobolev's inequalities for bounded domains and singular elliptic equations

“…For a proof of this fact see [23]. Typical examples of Young functionsB that satisfy (2.3) are given bȳ…”

Brezis–Marcus type inequalities with Lamb constant

“…is valid (see for example [15], [21] and [22]). The constant 1/4 in (1) is sharp for any convex subdomain of R n .…”