## Stability with explicit constants of the critical points of the fractional Sobolev inequality and applications to fast diffusion

created by denitti on 19 Nov 2022
modified on 22 Nov 2022

[BibTeX]

Submitted Paper

Inserted: 19 nov 2022
Last Updated: 22 nov 2022

Year: 2022

ArXiv: 2211.10634 PDF

Abstract:

We study the quantitative stability of critical points of the fractional Sobolev inequality. We show that, for a non-negative function $u \in \dot H^s(\mathbb R^N)$ whose energy satisfies $\tfrac{1}{2} S^\frac{N}{2s}_{N,s} \le \ u\ _{\dot H^s(\mathbb R^N)} \le \tfrac{3}{2}S_{N,s}^\frac{N}{2s},$ where $S_{N,s}$ is the optimal Sobolev constant, the bound $\ u -U[z,\lambda]\ _{\dot{H}^s(\mathbb R^N)} \lesssim \ (-\Delta)^s u - u^{2^*_s-1}\ _{\dot{H}^{-s}(\mathbb R^N)},$ holds for a suitable fractional Talenti bubble $U[z,\lambda]$. Special attention is paid to the explicit value of the implied constant in this stability inequality, which is new even for $s = 1$. As an application, we derive an explicit polynomial extinction rate for positive solutions to a fractional fast diffusion equation.