Sobolev inequalities for the Hardy-Schrödinger operator: Extremals and critical dimensions

2015/06/18 by Nassif Ghoussoub, Ghoussoub, Nassif, Frédéric Robert +1 · 1 citation
Computer Science · Mathematics · #35B44 #35J35 #35J60 #58J05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · doi:10.48550/arxiv.1506.05787

openalex publication_date 2015/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this expository paper, we consider the Hardy-Schrödinger operator -Δ-γ/|x|² on a smooth domain Ωof Rⁿ with 0∈Ω, and describe how the location of the singularity 0, be it in the interior of Ωor on its boundary, affects its analytical properties. We compare the two settings by considering the optimal Hardy, Sobolev, and the Caffarelli-Kohn-Nirenberg inequalities. The latter rewrites: C(∫_Ω\fracu^p|x|^sdx)^(2)/(p)≤ ∫_Ω |∇ u|²dx-γ∫_Ω(u²)/(|x|²)dx for all u∈ H¹₀(Ω), where γ

Citations

• On Hardy-Sobolev embedding
• Conformally flat manifolds, Kleinian groups and scalar curvature

Cited by

• A Nonlinear elliptic PDE with curve singularity on the boundary

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