Submitted Paper

Inserted: 19 nov 2022
Last Updated: 22 nov 2022

Year: 2022

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.