### 学术报告五十六：魏军城—Sharp quantitative estimates for Struwe’s decomposition

###### 时间：2021-07-05作者：点击数:_showDynClicks("wbnews", 1747959454, 4770)

Suppose $u\in \dot{H}^1(\mathbb{R}^n)$. In a seminal work, Struwe proved that if $u\geq 0$ and $\Gamma(u):=\|\Delta u+u^{\frac{n+2}{n-2}}\|_{H^{-1}}\to 0$ then $dist(u,\mathcal{T})\to0$,where$dist(u,\mathcal{T})$denotesthe$\dot{H}^1(\mathbb{R}^n)$-distance of $u$ from the manifold of sums of Talenti bubbles. Ciraolo, Figalli and Maggi obtained the first quantitative version of Struwe's decomposition with one bubble in all dimensions, namely $dist (u,\mathcal{T}) \leq C \Gamma (u)$. For Struwe's decomposition with two or more bubbles, Figalli and Glaudo showed a striking dimensional dependent quantitative estimate, namely $dist(u,\mathcal{T})\leq C \Gamma(u)$ when $3\leq n\leq 5$ while this is false for $n\geq 6$. In this talk, I will present the following sharp estimate

$dist(u,\mathcal{T})\leqC\begin{cases}\Gamma(u)\left|\log\Gamma(u)\right|^{\frac{1}{2}}\quad&\textit{if }n=6, |\Gamma(u)|^{\frac{n+2}{2(n-2)}}\quad&\textit{if }n\geq 7.\end{cases}$

Furthermore, we show that this inequality is sharp. (Joint work with B. Deng and L. Sun)