A note on the Lp-Sobolev inequality
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913574711984128 |
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| author | Deng, Shengbing Tian, Xingliang |
| author_facet | Deng, Shengbing Tian, Xingliang |
| contents | The usual Sobolev inequality in $\mathbb{R}^N$, asserts that $\|\nabla u\|_{L^p(\mathbb{R}^N)} \geq \mathcal{S}\|u\|_{L^{p^*}(\mathbb{R}^N)}$ for $1<p<N$ and $p^*=\frac{pN}{N-p}$, with $\mathcal{S}$ being the sharp constant. Based on a recent work of Figalli and Zhang [Duke Math. J., 2022], a weak norm remainder term of Sobolev inequality in a subdomain $Ω\subset \mathbb{R}^N$ with finite measure is established, i.e., for $\frac{2N}{N+1}<p<N$ there exists a constant $\mathcal{C}>0$ independent of $Ω$ such that \[ \|\nabla u\|^p_{L^p(Ω)}
-\mathcal{S}^p\|u\|^p_{L^{p^*}(Ω)} \geq \mathcal{C}|Ω|^{-\fracγ{p^*(p-1)}} \|u\|_{L^{\bar{p}}_w(Ω)}^γ\| u\|_{L^{p^*}(Ω)}^{p-γ},\quad \mbox{for all}\ u\in C^\infty_0(Ω)\setminus\{0\}, \] where $γ=\max\{2,p\}$, $\bar{p}=p^*(p-1)/p$, and $\|\cdot\|_{L^{\bar{p}}_w(Ω)}$ denotes the weak $L^{\bar{p}}$-norm. Moreover, we establish a sharp upper bound of Sobolev inequality in $\mathbb{R}^N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00464 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on the Lp-Sobolev inequality Deng, Shengbing Tian, Xingliang Analysis of PDEs The usual Sobolev inequality in $\mathbb{R}^N$, asserts that $\|\nabla u\|_{L^p(\mathbb{R}^N)} \geq \mathcal{S}\|u\|_{L^{p^*}(\mathbb{R}^N)}$ for $1<p<N$ and $p^*=\frac{pN}{N-p}$, with $\mathcal{S}$ being the sharp constant. Based on a recent work of Figalli and Zhang [Duke Math. J., 2022], a weak norm remainder term of Sobolev inequality in a subdomain $Ω\subset \mathbb{R}^N$ with finite measure is established, i.e., for $\frac{2N}{N+1}<p<N$ there exists a constant $\mathcal{C}>0$ independent of $Ω$ such that \[ \|\nabla u\|^p_{L^p(Ω)} -\mathcal{S}^p\|u\|^p_{L^{p^*}(Ω)} \geq \mathcal{C}|Ω|^{-\fracγ{p^*(p-1)}} \|u\|_{L^{\bar{p}}_w(Ω)}^γ\| u\|_{L^{p^*}(Ω)}^{p-γ},\quad \mbox{for all}\ u\in C^\infty_0(Ω)\setminus\{0\}, \] where $γ=\max\{2,p\}$, $\bar{p}=p^*(p-1)/p$, and $\|\cdot\|_{L^{\bar{p}}_w(Ω)}$ denotes the weak $L^{\bar{p}}$-norm. Moreover, we establish a sharp upper bound of Sobolev inequality in $\mathbb{R}^N$. |
| title | A note on the Lp-Sobolev inequality |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2401.00464 |