| Abstract: |
| We study quantitative stability for the classical Brezis-Nirenberg problem associated with the critical Sobolev embedding $H_0^1(\Omega) \hookrightarrow L^{\frac{2n}{n-2}}(\Omega)$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$ ($n \geq 3$). To the best of our knowledge, this is the first quantitative stability result for the Sobolev inequality on bounded domains. A key discovery of our work is the emergence of unexpected stability exponents, arising from an intricate interaction among the nonnegative solution $u_0$, the lower-order term $\lambda u$ in the Brezis-Nirenberg equation, bubble formation, and the boundary effect of the domain. A main difficulty is to quantify this boundary effect, which makes the problem fundamentally different from the Euclidean setting and from the case of smooth closed manifolds. Our proof refines and streamlines several arguments from the existing literature, while also resolving new analytical difficulties specific to the bounded domains. This talk is based on joint work with Haixia Chen (Hanyang University, Central China Normal University) and Juncheng Wei (The Chinese University of Hong Kong). |
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