Calculus of Variations and Geometric Measure Theory

N. De Nitti - T. K├Ânig

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


Submitted Paper

Inserted: 19 nov 2022
Last Updated: 22 nov 2022

Year: 2022

ArXiv: 2211.10634 PDF


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 \[ \tfrac{1}{2} S^\frac{N}{2s}_{N,s} \le \
_{\dot H^s(\mathbb R^N)} \le \tfrac{3}{2}S_{N,s}^\frac{N}{2s}, \] 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.