Sharp quantitative estimates of Struwe's Decomposition

par Bin Deng (IMT)

lundi 3 juin 2024, 14:00 → 15:00 Europe/Paris

Salle Pellos (1R2-207)

In a seminal work,  Struwe proved that if $0\le u\in {\dot{H}}^1({\mathbb{R}}^n)$ and $\mathrm{\Gamma }(u):=\Vert \mathrm{\Delta }u+u^{\frac{n+2}{n-2}}{\Vert }_{H^{-1}}\to 0$ then $dist(u,\mathcal{T})\to 0$, where $\mathcal{T}$ denotes the manifold of sums of Aubin-Talenti bubbles and $dist(u,\mathcal{T})$ denotes the ${\dot{H}}^1({\mathbb{R}}^n)$-distance of $u$ from $\mathcal{T}$.  Ciraolo, Figalli and Maggi obtained the first quantitative version of Struwe's decomposition with one bubble in all dimensions, namely $dist(u,\mathcal{T})\le C\mathrm{\Gamma }(u)$. For two or more bubbles, Figalli and Glaudo showed a striking dimensional dependent quantitative estimate, namely  $dist(u,\mathcal{T})\le C\mathrm{\Gamma }(u)$ when $3\le n\le 5$ while this is false for $n\ge 6$. In this talk, I will first show how to get a quantitative estimate, essentially a nonlinear inequality in higher dimensions. Afterward, I will show that this inequality is sharp by constructing an example. This is a joint work with Liming Sun (AMSS) and Juncheng Wei (UBC/CUHK).