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

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}{πd e}|\nabla u|{L^2($\mathbb{R}$^d)}^2\bigg), \end{eqnarray*} where $u\in W^{1,2}({\mathbb{R}}^d)$ with $d\ge 2$ and $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2({\mathbb{R}}^d)}=1$. It is well known that extremal functions of this inequality are precisely the Gaussians \begin{eqnarray*} \mathfrak{g}{σ,z}(x)=(πσ)^{-\frac{d}{2}}\mathfrak{g}{*}\bigg(\sqrt{\fracσ{2}}(x-z)\bigg)\quad$\text{with}$\quad \mathfrak{g}{}(x)=e^{-\frac{|x|^2}{2}}. \end{eqnarray} We prove that if $u\ge 0$ satisfying $(\nu -\frac{1}{2})c_0<\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1({\mathbb{R}}^d)}^2<(\nu +\frac{1}{2})c_0$ and $\mathrm{\parallel }-\mathrm{\Delta }u+u-2u\log \mathrm{\mid }u\mathrm{\mid }{\mathrm{\parallel }}_{H^{-1}}\le \delta$, where $c_0=\mathrm{\parallel }{\mathfrak{g}}_{1,0}{\mathrm{\parallel }}_{H^1({\mathbb{R}}^d)}^2$, $\nu \in \mathbb{N}$ and $\delta >0$ sufficiently small, then \begin{eqnarray*} $\text{dist}${H^1}(u, $\mathcal{M}$^ν)\lesssim|-Δu+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*} $\mathcal{M}$^ν={(\mathfrak{g}{1,0}(\cdot-z_1), \mathfrak{g}{1,0}(\cdot-z_2), \cdots, \mathfrak{g}_{1,0}(\cdot-z_ν))\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.