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}{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 $\Vert u{\Vert }_{L^2({\mathbb{R}}^d)}=1$. 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 $(\nu -\frac{1}{2})c_0<\Vert u{\Vert }_{H^1({\mathbb{R}}^d)}^2<(\nu +\frac{1}{2})c_0$ and $\Vert -\Delta u+u-2u\log |u|{\Vert }_{H^{-1}}\le \delta$, where $c_0=\Vert {\mathfrak{g}}_{1,0}{\Vert }_{H^1({\mathbb{R}}^d)}^2$, $\nu \in \mathbb{N}$ 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.

22 pages; contribution to Special Volume of AAG on Potentials & PDEs in memory of David R. Adams