2022/07/22 by Mousomi Bhakta, Bhakta, Mousomi, Debdip Ganguly +5 · 1 citation
Mathematics · #35A23 #35J61 #46E35 #58J05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2207.11024
openalex publication_date 2022/07/22 · openalex created_date 2022/07/27 · openalex updated_date 2026/07/28
Consider the Poincaré-Sobolev inequality on the hyperbolic space: for every n ≥ 3 and 1 < p ≤ (n+2)/(n-2), there exists a best constant Sn,p, λ(\mathbbBn)>0 such that Sn, p, λ(\mathbbBn)(~∫ _\mathbbBn|u|p+1 \rm dv_\mathbbBn )(2)/(p+1) ≤∫ _\mathbbBn(|∇_\mathbbBnu|2-λu2) \rm dv_\mathbbBn, holds for all u∈ Cc∞(\mathbbBn), and λ≤ ((n-1)2)/(4), where ((n-1)2)/(4) is the bottom of the L2-spectrum of -Δ_\mathbbBn. It is known from the results of Mancini and Sandeep [Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008)] that under appropriate assumptions on n,p and λ there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant Sn,p,λ(\mathbbBn). In this article, we investigate the quantitative gradient stability of the above inequality and the corresponding Euler-Lagrange equation locally around a bubble. Our result generalizes the sharp quantitative stability of Sobolev inequality in ℝn of Bianchi-Egnell [J. Funct. Anal. 100 (1991)] and Ciraolo-Figalli-Maggi [Int. Math. Res. Not. IMRN 2018] to the Poincaré-Sobolev inequality on the hyperbolic space. Furthermore, combining our stability results and implementing a refined smoothing estimates, we prove a quantitative extinction rate towards its basin of attraction of the solutions of the sub-critical fast diffusion flow for radial initial data. In another application, we derive sharp quantitative stability of the Hardy-Sobolev-Maz'ya inequalities for the class of functions which are symmetric in the component of singularity.