论文标题
双曲线空间中的庞贝 - 索伯夫不平等的急剧定量稳定性和快速扩散流的应用
Sharp quantitative stability of Poincare-Sobolev inequality in the hyperbolic space and applications to fast diffusion flows
论文作者
论文摘要
考虑双曲线空间上的poincaré-sobolev不平等：对于每$ n \ geq 3 $和$ 1 <p \ leq \ frac {n+2} {n-2} {n-2} {n-2} {n-2}，存在最佳常数$ s_ {n，p，λ}（\ nip，λ}（ λ}（\ Mathbb {b}^{n}）\ left（〜\ \ int \ limits _ {\ Mathbb {\ Mathbb {b}^{n}}} | U |^{p+1} \，{\ rm d} v _ {\ rm d} v _ { \ right）^{\ frac {2} {p+1}}} \ leq \ int \ limits _ {\ mathbb {b}^{n}}} \ left（| \ nabla _ {\ Mathbb {\ Mathbb {b}^{n}}} d} v _ {\ mathbb {b}^n}，$$在c_c^{\ infty}（\ mathbb {b}^n），$ u \ in y y y \ in y y \ y \ y \ u \中都保留，$ u \ {b}^n），$和$λ\ leq \ leq \ leq \ frac {（n-1）^2） $ l^2 $ -spectrum的$-Δ_ {\ Mathbb {b}^n}。$从Mancini和Sandeep的结果中知道。 sc。规范。极好的。比萨CL。科学。 7 (2008)] that under appropriate assumptions on $n,p$ and $λ$ there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant $S_{n,p,λ}(\mathbb{B}^n).$ In this article, we investigate the quantitative gradient stability of the above inequality and the corresponding Euler-Lagrange equation locally around a bubble. 我们的结果概括了Bianchi-Egnell的$ \ Mathbb {r}^n $中Sobolev不平等的急剧定量稳定性[J.功能。肛门。 100（1991）]和ciraolo-figalli-maggi [int。数学。 res。不是。 IMRN 2018]到圆环空间上的Poincaré-Sobolev不平等。 此外，将我们的稳定性结果结合在一起并实施精致的平滑估计值，我们证明了对径向初始数据的亚临界快速扩散流的吸引盆地的底层灭绝率。在另一个应用程序中，我们得出了对奇异性函数类别的Hardy-Sobolev-Maz'ya不平等现象的急剧定量稳定性。
Consider the Poincaré-Sobolev inequality on the hyperbolic space: for every $n \geq 3$ and $1 < p \leq \frac{n+2}{n-2},$ there exists a best constant $S_{n,p, λ}(\mathbb{B}^{n})>0$ such that $$S_{n, p, λ}(\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}-λu^{2}\right) \, {\rm d}v_{\mathbb{B}^n},$$ holds for all $u\in C_c^{\infty}(\mathbb{B}^n),$ and $λ\leq \frac{(n-1)^2}{4},$ where $\frac{(n-1)^2}{4}$ is the bottom of the $L^2$-spectrum of $-Δ_{\mathbb{B}^n}.$ 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 $λ$ there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant $S_{n,p,λ}(\mathbb{B}^n).$ 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 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.
