A geometrical approach to the sharp Hardy inequality in Sobolev-Slobodecki\uı spaces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911182851407872 |
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| author | Bianchi, Francesca Stefani, Giorgio Zagati, Anna Chiara |
| author_facet | Bianchi, Francesca Stefani, Giorgio Zagati, Anna Chiara |
| contents | We give a partial negative answer to a question left open in a previous work by Brasco and the first and third-named authors concerning the sharp constant in the fractional Hardy inequality on convex sets. Our approach has a geometrical flavor and equivalently reformulates the sharp constant in the limit case $p=1$ as the Cheeger constant for the fractional perimeter and the Lebesgue measure with a suitable weight. As a by-product, we obtain new lower bounds on the sharp constant in the $1$-dimensional case, even for non-convex sets, some of which optimal in the case $p=1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_08373 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A geometrical approach to the sharp Hardy inequality in Sobolev-Slobodecki\uı spaces Bianchi, Francesca Stefani, Giorgio Zagati, Anna Chiara Analysis of PDEs Functional Analysis Primary 49Q20. Secondary 35R11, 46E35 We give a partial negative answer to a question left open in a previous work by Brasco and the first and third-named authors concerning the sharp constant in the fractional Hardy inequality on convex sets. Our approach has a geometrical flavor and equivalently reformulates the sharp constant in the limit case $p=1$ as the Cheeger constant for the fractional perimeter and the Lebesgue measure with a suitable weight. As a by-product, we obtain new lower bounds on the sharp constant in the $1$-dimensional case, even for non-convex sets, some of which optimal in the case $p=1$. |
| title | A geometrical approach to the sharp Hardy inequality in Sobolev-Slobodecki\uı spaces |
| topic | Analysis of PDEs Functional Analysis Primary 49Q20. Secondary 35R11, 46E35 |
| url | https://arxiv.org/abs/2407.08373 |