On critical dipoles in dimensions n≥ 3

2021/01/23 by S. Blake Allan, Fritz Gesztesy, Allan, S. Blake +1
Computer Science · Mathematics · #35J30 #47F05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Primary: 35A23 #Secondary: 47A63 #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2101.09457

openalex publication_date 2021/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We reconsider generalizations of Hardy's inequality corresponding to the case of (point) dipole potentials V_γ(x) = γ(u, x) |x|^-3, x ∈ ℝⁿ \backslash \0\, γ∈ [0,∞), u ∈ ℝⁿ, |u|=1, n ∈ ℕ, n ≥ 3. More precisely, for n ≥ 3, we provide an alternative proof of the existence of a critical dipole coupling constant γ_c,n > 0, such that amp;\textfor all γ∈ [0,γ_c,n], and all u ∈ ℝⁿ, |u|=1,
amp; ∫_ℝⁿ dⁿ x |(∇ f)(x)|² ≥ ± γ∫_ℝⁿ dⁿ x (u, x) |x|^-3 |f(x)|², f ∈ D¹(ℝⁿ). with D¹(ℝⁿ) denoting the completion of C₀^∞(ℝⁿ) with respect to the norm induced by the gradient. Here γ_c,n is sharp, that is, the largest possible such constant, and we discuss a numerical scheme for its computation. Moreover, we discuss upper and lower bounds for γ_c,n > 0. We also consider the case of multicenter dipole interactions with dipoles centered on an infinite discrete set.

Citations

• On the Non-Existence of Conjugate Points
• Decoupling of deficiency indices and applications to Schrödinger-type operators with possibly strongly singular potentials
• A Derivation of n-Dimensional Spherical Coordinates

Related

• Critical Hardy--Sobolev Inequalities
• On the best constant of Hardy-Sobolev Inequalities
• Multipolar Hardy inequalities and mutual interaction of the poles
• Hardy's inequality in a limiting case on general bounded domains
• Sharp numerical approximation of the Hardy constant