| 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, 2007) 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 talk we will discuss the quantitative gradient stability of the above inequality and the associated Euler-Lagrange equation locally around a bubble.
We will show sharp quantitative stability of the above Poincar\`e-Sobolev inequality: if $u \in H^1(\mathbb{B}^n)$ almost optimizes the above inequality then $u$ is close to the manifold of optimizers in a quantitative way. Secondly, we will discuss
the quantitative stability of its critical points: if $u \in H^1({\mathbb{B}^n})$ almost solves the Euler-Lagrange equation corresponding to the above Poincar\`e-Sobolev inequality and the energy of $u$ is close to the energy of an extremizer, then the following quantitative bound holds
$$
\hbox{dist}\left(u, \mathcal{Z}\right) \leq C(n,p,\lambda) \|\Delta_{\mathbb{B}^n} u + \lambda u + u^p \|_{H^{-1}(\mathbb{B}^n)},
$$
where $\mathcal{Z}$ denotes the manifold of non-negative finite energy solutions of
$ -\Delta_{\mathbb{B}^{n}}w-\lambda w=|w|^{p-1}w$. Our result generalizes the sharp quantitative stability of
Sobolev inequality in Euclidean space of Bianchi-Egnell ((J. Funct. Anal. 1991) and Ciraolo-Figalli-Maggi (Int. Math.
Res. Not. IMRN, 2021) to the Poincar\`{e}-Sobolev
inequality on the hyperbolic space. |
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