Deep estimates for higher eigenvalues of the poly-Laplacian
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918135542579200 |
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| author | Ji, Zhengchao Xu, Hongwei |
| author_facet | Ji, Zhengchao Xu, Hongwei |
| contents | We investigate the lower bound for higher eigenvalues $λ_i$ of the poly-Laplace operator on a bounded domain and improve the famous Li-Yau inequality and its related results. Firstly, we consider the low dimensional cases for the Pólya conjecture, the clamped plate problem and the eigenvalue problem of the poly-Laplacian and deliver a series of deep eigenvalue inequalities for these problems respectively. Secondly, we establish a sharp lower bound for the eigenvalues of the poly-Laplacia in arbitrary dimension under some certain restrictive conditions. Finally, we provide an improved inequality for $λ_i$ in arbitrary dimension without any restrictive conditions. Our results also yield the improvement of the lower bounds for the Stokes eigenvalue problems and the Generalized Pólya conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_04069 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Deep estimates for higher eigenvalues of the poly-Laplacian Ji, Zhengchao Xu, Hongwei Differential Geometry We investigate the lower bound for higher eigenvalues $λ_i$ of the poly-Laplace operator on a bounded domain and improve the famous Li-Yau inequality and its related results. Firstly, we consider the low dimensional cases for the Pólya conjecture, the clamped plate problem and the eigenvalue problem of the poly-Laplacian and deliver a series of deep eigenvalue inequalities for these problems respectively. Secondly, we establish a sharp lower bound for the eigenvalues of the poly-Laplacia in arbitrary dimension under some certain restrictive conditions. Finally, we provide an improved inequality for $λ_i$ in arbitrary dimension without any restrictive conditions. Our results also yield the improvement of the lower bounds for the Stokes eigenvalue problems and the Generalized Pólya conjecture. |
| title | Deep estimates for higher eigenvalues of the poly-Laplacian |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2508.04069 |