Sobolev improvements on sharp Rellich inequalities
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913247452463104 |
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| author | Barbatis, Gerassimos Tertikas, Achilles |
| author_facet | Barbatis, Gerassimos Tertikas, Achilles |
| contents | There are two Rellich inequalities for the bilaplacian, that is for $\int (Δu)^2dx$, the one involving $|\nabla u|$ and the other involving $|u|$ at the RHS. In this article we consider these inequalities with sharp constants and obtain sharp Sobolev-type improvements. More precisely, in our first result we improve the Rellich inequality with $|\nabla u|$ obtained recently by Cazacu in dimensions $n=3,4$ by a sharp Sobolev term thus complementing existing results for the case $n\geq 5$. In the second theorem the sharp constant of the Sobolev improvement for the Rellich inequality with $|u|$ is obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_00433 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sobolev improvements on sharp Rellich inequalities Barbatis, Gerassimos Tertikas, Achilles Analysis of PDEs 35A23, 35G05, 35G20 (Primary) 31B30, 35J30, 46E35 (Secondary) There are two Rellich inequalities for the bilaplacian, that is for $\int (Δu)^2dx$, the one involving $|\nabla u|$ and the other involving $|u|$ at the RHS. In this article we consider these inequalities with sharp constants and obtain sharp Sobolev-type improvements. More precisely, in our first result we improve the Rellich inequality with $|\nabla u|$ obtained recently by Cazacu in dimensions $n=3,4$ by a sharp Sobolev term thus complementing existing results for the case $n\geq 5$. In the second theorem the sharp constant of the Sobolev improvement for the Rellich inequality with $|u|$ is obtained. |
| title | Sobolev improvements on sharp Rellich inequalities |
| topic | Analysis of PDEs 35A23, 35G05, 35G20 (Primary) 31B30, 35J30, 46E35 (Secondary) |
| url | https://arxiv.org/abs/2312.00433 |