On weighted L^p-Hardy inequality on domains in ℝⁿ

2020/12/23 by Divya Goel, Goel, Divya, Yehuda Pinchover +3
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2012.12860

Abstract

We consider weighted L^p-Hardy inequalities involving the distance to the boundary of a domain in the n-dimensional Euclidean space with nonempty boundary. Using criticality theory, we give an alternative proof of the following result of F.~G.~Avkhadiev (2006) Theorem: Let Ω\subsetneqq ℝⁿ, n≥ 2, be an arbitrary domain, 1n. Let d_Ω(x) =dist(x,∂ Ω) denote the distance of a point x∈ Ω to ∂ Ω. Then the following Hardy-type inequality holds ∫_Ω(|∇ φ|^p)/(d_Ω^α) dx ≥ ( (α+p-n)/(p))^p ∫_Ω\frac|φ|^pd_Ω^p+α dx ∀ φ∈ C^∞ _c(Ω), and the lower bound constant ( (α+p-n)/(p))^p is sharp.

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