On the Ashbaugh-Benguria type conjecture about lower-order Neumann eigenvalues of the Witten-Laplacian
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866908405274247168 |
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| author | Chen, Ruifeng Mao, Jing |
| author_facet | Chen, Ruifeng Mao, Jing |
| contents | An isoperimetric inequality for lower order nonzero Neumann eigenvalues of the Witten-Laplacian on bounded domains in a Euclidean space or a hyperbolic space has been proven in this paper. About this conclusion, we would like to point out two things:
It strengthens the well-known Szegő-Weinberger inequality for nonzero Neumann eigenvalues of the classical free membrane problem given in [J. Rational Mech. Anal. 3 (1954) 343-356] and [J. Rational Mech. Anal. 5 (1956) 633-636];
Recently, Xia-Wang [Math. Ann. 385 (2023) 863-879] gave a very important progress to the celebrated conjecture of M. S. Ashbaugh and R. D. Benguria proposed in [SIAM J. Math. Anal. 24 (1993) 557-570]. It is easy to see that our conclusion here covers Xia-Wang's this progress as a special case.
In this paper, we have also proposed two open problems which can be seen as a generalization of Ashbaugh-Benguria's conjecture mentioned above. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_08070 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Ashbaugh-Benguria type conjecture about lower-order Neumann eigenvalues of the Witten-Laplacian Chen, Ruifeng Mao, Jing Analysis of PDEs Differential Geometry 35P15, 49Jxx, 35J15 An isoperimetric inequality for lower order nonzero Neumann eigenvalues of the Witten-Laplacian on bounded domains in a Euclidean space or a hyperbolic space has been proven in this paper. About this conclusion, we would like to point out two things: It strengthens the well-known Szegő-Weinberger inequality for nonzero Neumann eigenvalues of the classical free membrane problem given in [J. Rational Mech. Anal. 3 (1954) 343-356] and [J. Rational Mech. Anal. 5 (1956) 633-636]; Recently, Xia-Wang [Math. Ann. 385 (2023) 863-879] gave a very important progress to the celebrated conjecture of M. S. Ashbaugh and R. D. Benguria proposed in [SIAM J. Math. Anal. 24 (1993) 557-570]. It is easy to see that our conclusion here covers Xia-Wang's this progress as a special case. In this paper, we have also proposed two open problems which can be seen as a generalization of Ashbaugh-Benguria's conjecture mentioned above. |
| title | On the Ashbaugh-Benguria type conjecture about lower-order Neumann eigenvalues of the Witten-Laplacian |
| topic | Analysis of PDEs Differential Geometry 35P15, 49Jxx, 35J15 |
| url | https://arxiv.org/abs/2403.08070 |