2007/09/12 by Nassif Ghoussoub, Nassif Ghoussouband, Amir Moradifam +2
Mathematics · #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics #math.AP
paper · pdf · doi:10.48550/arxiv.0709.1954
35 pages. Updated versions --if any-- of this author's papers can be downloaded at http://www.birs.ca/~nassif
arxiv created 2007/09/12 · openalex publication_date 2007/09/12 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in Rn,n ≥ 1, so that the following inequalities hold for all u ∈ C0∞(B): ∫BV(x)|∇ u |2dx ≥ ∫B W(x)u2dx, and ∫BV(x)|Δu |2dx ≥ ∫B W(x)|∇ u|2dx+(n-1)∫B((V(x))/(|x|2)-(Vr(|x|))/(|x|))|∇ u|2dx. This characterization makes a very useful connection between Hardy-type inequalities and the oscillatory behaviour of certain ordinary differential equations, and helps in the identification of a large number of such couples (V, W) - that we call Bessel pairs -as well as the best constants in the corresponding inequalities. This allows us to improve, extend, and unify many results -old and new- about Hardy and Hardy-Rellich type inequalities, such as those obtained by Caffarelli-Kohn-Nirenberg, Brezis-Vazquez, Wang-Willem, Adimurthi-Chaudhuri-Ramaswamy, Filippas-Tertikas, Adimurthi-Grossi -Santra, Tertikas-Zographopoulos, and Blanchet-Bonforte-Dolbeault-Grillo-Vasquez.