On fractional Hardy-type inequalities in general open sets
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909248016875520 |
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| author | Cinti, Eleonora Prinari, Francesca |
| author_facet | Cinti, Eleonora Prinari, Francesca |
| contents | We show that, when $sp>N$, the sharp Hardy constant $\mathfrak{h}_{s,p}$ of the punctured space $\mathbb R^N\setminus\{0\}$ in the Sobolev-Slobodecki\uı space provides an optimal lower bound for the Hardy constant $\mathfrak{h}_{s,p}(Ω)$ of an open $Ω\subsetneq \mathbb R^N$. The proof exploits the characterization of Hardy's inequality in the fractional setting in terms of positive local weak supersolutions of the relevant Euler-Lagrange equation and relies on the construction of suitable supersolutions by means of the distance function from the boundary of $Ω$. Moreover, we compute the limit of $\mathfrak{h}_{s,p}$ as $s\nearrow 1$, as well as the limit when $p \nearrow \infty$. Finally, we apply our results to establish a lower bound for the non-local eigenvalue $λ_{s,p}(Ω)$ in terms of $\mathfrak{h}_{s,p}$ when $sp>N$, which, in turn, gives an improved Cheeger inequality whose constant does not vanish as $p\nearrow \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06568 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On fractional Hardy-type inequalities in general open sets Cinti, Eleonora Prinari, Francesca Analysis of PDEs Spectral Theory 46E35, 39B72, 35R11 We show that, when $sp>N$, the sharp Hardy constant $\mathfrak{h}_{s,p}$ of the punctured space $\mathbb R^N\setminus\{0\}$ in the Sobolev-Slobodecki\uı space provides an optimal lower bound for the Hardy constant $\mathfrak{h}_{s,p}(Ω)$ of an open $Ω\subsetneq \mathbb R^N$. The proof exploits the characterization of Hardy's inequality in the fractional setting in terms of positive local weak supersolutions of the relevant Euler-Lagrange equation and relies on the construction of suitable supersolutions by means of the distance function from the boundary of $Ω$. Moreover, we compute the limit of $\mathfrak{h}_{s,p}$ as $s\nearrow 1$, as well as the limit when $p \nearrow \infty$. Finally, we apply our results to establish a lower bound for the non-local eigenvalue $λ_{s,p}(Ω)$ in terms of $\mathfrak{h}_{s,p}$ when $sp>N$, which, in turn, gives an improved Cheeger inequality whose constant does not vanish as $p\nearrow \infty$. |
| title | On fractional Hardy-type inequalities in general open sets |
| topic | Analysis of PDEs Spectral Theory 46E35, 39B72, 35R11 |
| url | https://arxiv.org/abs/2407.06568 |