Sharp stability of the logarithmic Sobolev inequality in the critical point setting

Abstract: In this paper, we consider the Euclidean logarithmic Sobolev inequality \begin{eqnarray*} \int_{\mathbb{R}^d}|u|^2\log|u|dx\leq\frac{d}{4}\log\bigg(\frac{2}{\pi d e}|\nabla u|{L^2(\mathbb{R}^d)}^2\bigg), \end{eqnarray*} where $u\in W^{1,2}(\mathbb{R}^d)$ with $d\geq2$ and $|u|{L^2(\mathbb{R}^d)}=1$. It is well known that extremal functions of this inequality are precisely the Gaussians \begin{eqnarray*} \mathfrak{g}{\sigma,z}(x)=(\pi\sigma)^{-\frac{d}{2}}\mathfrak{g}{}\bigg(\sqrt{\frac{\sigma}{2}}(x-z)\bigg)\quad\text{with}\quad \mathfrak{g}_{}(x)=e^{-\frac{|x|^2}{2}}. \end{eqnarray*} We prove that if $u\geq0$ satisfying $(\nu-\frac12)c_0<|u|{H^1(\mathbb{R}^d)}^2<(\nu+\frac12)c_0$ and $|-\Delta u+u-2u\log |u||{H^{-1}}\leq\delta$, where $c_0=|\mathfrak{g}{1,0}|{H^1(\mathbb{R}^d)}^2$, $\nu\in \mathbb{N}$ and $\delta>0$ sufficiently small, then \begin{eqnarray*} \text{dist}{H^1}(u, \mathcal{M}^\nu)\lesssim|-\Delta u+u-2u\log |u||{H^{-1}} \end{eqnarray*} which is optimal in the sense that the order of the right hand side is sharp, where \begin{eqnarray*} \mathcal{M}^\nu={(\mathfrak{g}_{1,0}(\cdot-z_1), \mathfrak{g}{1,0}(\cdot-z_2), \cdots, \mathfrak{g}{1,0}(\cdot-z_\nu))\mid z_i\in\bbr^d}. \end{eqnarray*} Our result provides an optimal stability of the Euclidean logarithmic Sobolev inequality in the critical point setting.