论文标题
对数Sobolev不平等的急剧稳定性在临界点环境中
Sharp stability of the logarithmic Sobolev inequality in the critical point setting
论文作者
论文摘要
在本文中，我们考虑欧几里得对数sobolev不等式\ begin {eqnarray*} \ int _ {\ mathbb {\ mathbb {r}^d} | u | u |^2 \ log | u | u | dx \ leq \ leq \ leq \ frac \ frac \ frac {d} {d} {4} {4} {4} {4 e} \ | \ | \ nabla u \ | _ {l^2（\ mathbb {r}^d）}^2 \ bigg），\ end {eqnarray*}其中$ u \ in w^{1,2}（\ Mathbb {\ Mathb {r}^d）in $ \ | u \ | _ {l^2（\ Mathbb {r}^d）} = 1 $。众所周知，这种不平等的极端功能恰恰是高斯人\ begin {eqnarray*} \ Mathfrak {g} _ {σ，z}（x）=（πσ）^{ - \ frac {d} {2}} {2}} \ Mathfrak {g} _ {*} \ bigG（\ sqrt {\ sqrt {\fracσ}} {2}} {2}} {{2}} {x-Z） \ Mathfrak {g} _ {*}（x）= e^{ - \ frac {| x |^2} {2}}}。 \ end {eqnArray*}我们证明，如果$ u \ geq0 $满足$（ν-\ frac12）C_0 <\ | U \ | | _ { | u | \ | _ {h^{ - 1}} \leqδ$，其中$ c_0 = \ | \ | \ m rathfrak {g} _ {1,0} \ | _ {h^1（\ mathbb {\ mathbb {r}^d）} \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），\ cdot-Z_1），\ Mathfrak {g} _ {1,0} \ Mathfrak {g} _ {1,0}（\ cdot-Z_ν））\ Mid Z_i \ in \ bbr^d \}。 \ end {eqnarray*}我们的结果提供了在临界点设置中欧几里得对数sobolev不等式的最佳稳定性。
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 $d\geq2$ 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}_{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 $u\geq0$ satisfying $(ν-\frac12)c_0<\|u\|_{H^1(\mathbb{R}^d)}^2<(ν+\frac12)c_0$ and $\|-Δu+u-2u\log |u|\|_{H^{-1}}\leqδ$, where $c_0=\|\mathfrak{g}_{1,0}\|_{H^1(\mathbb{R}^d)}^2$, $ν\in \mathbb{N}$ and $δ>0$ 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.
