Sharp quantitative stability of Poincare-Sobolev inequality in the hyperbolic space and applications to fast diffusion flows

TL;DR

This paper establishes sharp quantitative stability results for the Poincaré-Sobolev inequality in hyperbolic space, extending classical Euclidean results and applying them to analyze fast diffusion flows and Hardy-Sobolev-Maz'ya inequalities.

Contribution

It generalizes the stability of Sobolev inequalities to hyperbolic space and applies these results to fast diffusion flows and Hardy-Sobolev-Maz'ya inequalities.

Findings

Quantitative gradient stability around bubbles in hyperbolic space.

Sharp extinction rates for fast diffusion flows with radial initial data.

Extension of Euclidean stability results to hyperbolic geometry.

Abstract

Consider the Poincar\'e-Sobolev inequality on the hyperbolic space: for every $n \geq 3$ and $1 < p \leq \frac{n+2}{n-2},$ there exists a best constant $S_{n,p, \lambda}(\mathbb{B}^{n})>0$ such that $$S_{n, p, \lambda}(\mathbb{B}^{n})\left(~\int \limits_{\mathbb{B}^{n}}|u|^{{p+1}} \, {\rm d}v_{\mathbb{B}^n} \right)^{\frac{2}{p+1}} \leq\int \limits_{\mathbb{B}^{n}}\left(|\nabla_{\mathbb{B}^{n}}u|^{2}-\lambda u^{2}\right) \, {\rm d}v_{\mathbb{B}^n},$$ holds for all $u\in C_c^{\infty}(\mathbb{B}^n),$ and $\lambda \leq \frac{(n-1)^2}{4},$ where $\frac{(n-1)^2}{4}$ is the bottom of the $L^2$-spectrum of $-\Delta_{\mathbb{B}^n}.$ 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 $\lambda$ there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant $S_{n,p,\lambda}(\mathbb{B}^n).$ 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 $\mathbb{R}^n$ of Bianchi-Egnell [J. Funct. Anal. 100 (1991)] and Ciraolo-Figalli-Maggi [Int. Math. Res. Not. IMRN 2018] to the Poincar\'{e}-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.