Report time: 9:30-10:30, July 09, 2021 (Friday)

Venue: Tencent Conference ID: 780 972 775

Speaker: Professor Wei Juncheng

Employer: University of British Columbia

Organized by: School of Mathematics

Introduction to the Report:

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).

Brief Introduction of speaker:

Wei Juncheng, a famous Chinese mathematician, is a member of the Royal Canadian Academy of Sciences. He received his BACHELOR's degree from Wuhan University in 1989 and was selected to study for his doctorate in the United States under the National Chen Shianxin Scholarship Program. He received his doctorate from the University of Minnesota in 1994. Currently, he holds the Canada Research Chair at the University of British Columbia. His research interests include nonlinear partial differential equations, condensation and blasting, and mathematical biology. He was awarded the Croucher Foundation of Hong Kong in 2005 In 2010, he received the Morningside Silver Award of Chinese Mathematical Congress, the First Prize of Natural Science of the Ministry of Education in 2010, was invited to give a 45-minute lecture at the 27th International Congress of Mathematicians in 2014, and received the Jeffery-Williams Award of Canadian Mathematical Society in 2020. Professor Wei juncheng has internationally recognized achievements in the fields of nonlinear partial differential equations and biomathetics. He has published more than 450 papers in international mathematical journals, including four top journals in mathematics, Ann. Math. And Invents. Math.