Sharp spectral gap estimates for higher-order operators on Cartan-Hadamard manifolds
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917629172645888 |
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| author | Farkas, Csaba Kajántó, Sándor Kristály, Alexandru |
| author_facet | Farkas, Csaba Kajántó, Sándor Kristály, Alexandru |
| contents | The goal of this paper is to provide sharp spectral gap estimates for problems involving higher-order operators (including both the clamped and buckling plate problems) on Cartan-Hadamard manifolds. The proofs are symmetrization-free -- thus no sharp isoperimetric inequality is needed -- based on two general, yet elementary functional inequalities. The spectral gap estimate for clamped plates solves a sharp asymptotic problem from Cheng and Yang [Proc. Amer. Math. Soc., 2011] concerning the behavior of higher-order eigenvalues on hyperbolic spaces, and answers a question raised in Kristály [Adv. Math., 2020] on the validity of such sharp estimates in high-dimensional Cartan-Hadamard manifolds. As a byproduct of the general functional inequalities, various Rellich inequalities are established in the same geometric setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_02911 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sharp spectral gap estimates for higher-order operators on Cartan-Hadamard manifolds Farkas, Csaba Kajántó, Sándor Kristály, Alexandru Analysis of PDEs Differential Geometry Functional Analysis The goal of this paper is to provide sharp spectral gap estimates for problems involving higher-order operators (including both the clamped and buckling plate problems) on Cartan-Hadamard manifolds. The proofs are symmetrization-free -- thus no sharp isoperimetric inequality is needed -- based on two general, yet elementary functional inequalities. The spectral gap estimate for clamped plates solves a sharp asymptotic problem from Cheng and Yang [Proc. Amer. Math. Soc., 2011] concerning the behavior of higher-order eigenvalues on hyperbolic spaces, and answers a question raised in Kristály [Adv. Math., 2020] on the validity of such sharp estimates in high-dimensional Cartan-Hadamard manifolds. As a byproduct of the general functional inequalities, various Rellich inequalities are established in the same geometric setting. |
| title | Sharp spectral gap estimates for higher-order operators on Cartan-Hadamard manifolds |
| topic | Analysis of PDEs Differential Geometry Functional Analysis |
| url | https://arxiv.org/abs/2307.02911 |