Sharp quantitative stability estimates for the Brezis-Nirenberg problem
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arXiv
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| Format: | Preprint |
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2025
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| author | Chen, Haixia Kim, Seunghyeok Wei, Juncheng |
| author_facet | Chen, Haixia Kim, Seunghyeok Wei, Juncheng |
| contents | We study the quantitative stability for the classical Brezis-Nirenberg problem associated with the critical Sobolev embedding $H^1_0(Ω) \hookrightarrow L^{\frac{2n}{n-2}}(Ω)$ in a smooth bounded domain $Ω\subset \mathbb{R}^n$ ($n \geq 3$). To the best of our knowledge, this work presents the first quantitative stability result for the Sobolev inequality on bounded domains. A key discovery is the emergence of unexpected stability exponents in our estimates, which arise from the intricate interaction among the nonnegative solution $u_0$ and the linear term $λu$ of the Brezis--Nirenberg equation, bubble formation, and the boundary effect of the domain $Ω$. One of the main challenges is to capture the boundary effect quantitatively, a feature that fundamentally distinguishes our setting from the Euclidean case treated in \cite{CFM, FG, DSW} and the smooth closed manifold case studied in \cite{CK}. In addressing a variety of difficulties, our proof refines and streamlines several arguments from the existing literature while also resolving new analytical challenges specific to our setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_07602 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp quantitative stability estimates for the Brezis-Nirenberg problem Chen, Haixia Kim, Seunghyeok Wei, Juncheng Analysis of PDEs We study the quantitative stability for the classical Brezis-Nirenberg problem associated with the critical Sobolev embedding $H^1_0(Ω) \hookrightarrow L^{\frac{2n}{n-2}}(Ω)$ in a smooth bounded domain $Ω\subset \mathbb{R}^n$ ($n \geq 3$). To the best of our knowledge, this work presents the first quantitative stability result for the Sobolev inequality on bounded domains. A key discovery is the emergence of unexpected stability exponents in our estimates, which arise from the intricate interaction among the nonnegative solution $u_0$ and the linear term $λu$ of the Brezis--Nirenberg equation, bubble formation, and the boundary effect of the domain $Ω$. One of the main challenges is to capture the boundary effect quantitatively, a feature that fundamentally distinguishes our setting from the Euclidean case treated in \cite{CFM, FG, DSW} and the smooth closed manifold case studied in \cite{CK}. In addressing a variety of difficulties, our proof refines and streamlines several arguments from the existing literature while also resolving new analytical challenges specific to our setting. |
| title | Sharp quantitative stability estimates for the Brezis-Nirenberg problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.07602 |