Higher order Hardy-Rellich identities
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917779823656960 |
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| author | Huang, Xia Ye, Dong |
| author_facet | Huang, Xia Ye, Dong |
| contents | In this paper, we show Hardy-Rellich identities for polyharmonic operators $Δ^m$ and radial Laplacian $Δ_r^m$ in $\mathbb{R}^n$ with Hardy-Hénon weight $|x|^α$ for all $m, n\in \mathbb{N}, α\in \mathbb{R}$. Moreover, the iterative method is applied to give Hardy-Rellich equalities with general weights on Riemannian manifolds. These identities provide naturally an alternative approach to obtain and improve Hardy-Rellich type inequalities. As example of application, we extend several Rellich inequalities of Tertikas-Zographopoulos (Adv. Math. 2007) to the weighted case; using equality with weights involving logarithmic, we show another new weighted Rellich estimate between integrals of $Δu$ and $|\nabla u|$; we establish also a Rellich identity involving the Laplace-Beltrami operator $Δ_\mathbb{H}$ and the radial Laplacian $Δ_{ρ, \mathbb{H}}$ of the hyperbolic space $\mathbb{H}^n$, which yields in particular brand-new Rellich inequalities for $\|Δ_\mathbb{H} u\|$ in $\mathbb{H}^3$ and $\mathbb{H}^4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12571 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher order Hardy-Rellich identities Huang, Xia Ye, Dong Analysis of PDEs 26D10 26D15 35A23 In this paper, we show Hardy-Rellich identities for polyharmonic operators $Δ^m$ and radial Laplacian $Δ_r^m$ in $\mathbb{R}^n$ with Hardy-Hénon weight $|x|^α$ for all $m, n\in \mathbb{N}, α\in \mathbb{R}$. Moreover, the iterative method is applied to give Hardy-Rellich equalities with general weights on Riemannian manifolds. These identities provide naturally an alternative approach to obtain and improve Hardy-Rellich type inequalities. As example of application, we extend several Rellich inequalities of Tertikas-Zographopoulos (Adv. Math. 2007) to the weighted case; using equality with weights involving logarithmic, we show another new weighted Rellich estimate between integrals of $Δu$ and $|\nabla u|$; we establish also a Rellich identity involving the Laplace-Beltrami operator $Δ_\mathbb{H}$ and the radial Laplacian $Δ_{ρ, \mathbb{H}}$ of the hyperbolic space $\mathbb{H}^n$, which yields in particular brand-new Rellich inequalities for $\|Δ_\mathbb{H} u\|$ in $\mathbb{H}^3$ and $\mathbb{H}^4$. |
| title | Higher order Hardy-Rellich identities |
| topic | Analysis of PDEs 26D10 26D15 35A23 |
| url | https://arxiv.org/abs/2409.12571 |