Hardy's inequality and curvature

2011/01/12 by Alexander Balinsky, Balinsky, A. A., W. D. Evans +3 · 1 citation
Mathematics · #Analytic and geometric function theory #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1101.2331

openalex publication_date 2011/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A Hardy inequality of the form ∫_Ω |∇ f(\bfx)|^p d \bfx ≥ ((p-1)/(p))^p ∫_Ω \1 + a(δ, ∂ Ω)(\x)\\frac|f(\bfx)|^pδ(\bfx)^p d\bfx, for all f ∈ C₀^∞(Ω), is considered for p∈ (1,∞), where Ω can be either Ω or ℝⁿ ∖ Ω with Ω a domain in ℝⁿ, n ≥ 2, and δ(\bfx) is the distance from \bfx ∈ Ω to the boundary ∂ Ω. The main emphasis is on determining the dependance of a(δ, ∂ Ω) on the geometric properties of ∂ Ω. A Hardy inequality is also established for any doubly connected domain Ω in ℝ² in terms of a uniformisation of Ω, that is, any conformal univalent map of Ω onto an annulus.

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• Sharp second order inequalities with distance function to the boundary and applications to a p-Biharmonic singular problem

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