Bivariate Hardy-Sobolev Inequality and Its Sharp Stability
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914303943114752 |
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| author | Zhang, Yingfang Zhong, Xuexiu Zou, Wenming |
| author_facet | Zhang, Yingfang Zhong, Xuexiu Zou, Wenming |
| contents | This paper establishes a bivariate Hardy-Sobolev inequality. Let $Ω\subset \mathbb{R}^N$ ($N \geq 3$) be an open domain, $s \in (0,2)$, $α> 1$, $β> 1$ with $α+ β= 2^*(s)$, and $κ\in \mathbb{R}$. For any functions $u, v \in D_0^{1,2}(Ω)$, we prove the inequality:
\begin{multline*}
\int_Ω |\nabla u|^2 \, \mathrm{d}x + \int_Ω |\nabla v|^2 \, \mathrm{d}x
\ge S_{α,β,λ,μ}(Ω) \left( \int_Ω \Big( λ\frac{|u|^{2^*(s)}}{|x|^s} + μ\frac{|v|^{2^*(s)}}{|x|^s} + 2^*(s) κ\frac{|u|^α|v|^β}{|x|^s} \Big)\, \mathrm{d}x \right)^{\frac{2}{2^*(s)}}.
\end{multline*}
We derive the best constant $S_{α,β,λ,μ}(Ω)$ and characterize the set of minimizers. Moreover, for $Ω= \mathbb{R}^N$ and $κ> 0$, we obtain sharp stability results for nonnegative functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03191 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bivariate Hardy-Sobolev Inequality and Its Sharp Stability Zhang, Yingfang Zhong, Xuexiu Zou, Wenming Analysis of PDEs This paper establishes a bivariate Hardy-Sobolev inequality. Let $Ω\subset \mathbb{R}^N$ ($N \geq 3$) be an open domain, $s \in (0,2)$, $α> 1$, $β> 1$ with $α+ β= 2^*(s)$, and $κ\in \mathbb{R}$. For any functions $u, v \in D_0^{1,2}(Ω)$, we prove the inequality: \begin{multline*} \int_Ω |\nabla u|^2 \, \mathrm{d}x + \int_Ω |\nabla v|^2 \, \mathrm{d}x \ge S_{α,β,λ,μ}(Ω) \left( \int_Ω \Big( λ\frac{|u|^{2^*(s)}}{|x|^s} + μ\frac{|v|^{2^*(s)}}{|x|^s} + 2^*(s) κ\frac{|u|^α|v|^β}{|x|^s} \Big)\, \mathrm{d}x \right)^{\frac{2}{2^*(s)}}. \end{multline*} We derive the best constant $S_{α,β,λ,μ}(Ω)$ and characterize the set of minimizers. Moreover, for $Ω= \mathbb{R}^N$ and $κ> 0$, we obtain sharp stability results for nonnegative functions. |
| title | Bivariate Hardy-Sobolev Inequality and Its Sharp Stability |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.03191 |