关于Witten-Laplacian低阶Neumann特征值的Ashbaugh-Benguria型猜想
On the Ashbaugh-Benguria type conjecture about lower-order Neumann eigenvalues of the Witten-Laplacian
AI总结：
本文证明欧氏与双曲空间中有界域上Witten-Laplacian低阶非零Neumann特征值的等周不等式，强化了Szegő-Weinberger不等式，涵盖Xia-Wang关于Ashbaugh-Benguria猜想的进展，并提出两个推广该猜想的开放问题。
摘要：
本文证明了欧氏空间或双曲空间中有界域上Witten-Laplacian的低阶非零Neumann特征值的等周不等式。关于该结论，我们指出两点：其一，它强化了经典自由膜问题非零Neumann特征值的著名Szegő-Weinberger不等式，该不等式最初见于《J. Rational Mech. Anal. 3 (1954) 343-356》与《J. Rational Mech. Anal. 5 (1956) 633-636》；其二，近期Xia-Wang在《Math. Ann. 385 (2023) 863-879》中针对M. S. Ashbaugh与R. D. Benguria在《SIAM J. Math. Anal. 24 (1993) 557-570》中提出的著名猜想取得了重要进展，而本文的结论可将Xia-Wang的这一进展作为特例涵盖。此外，本文还提出了两个开放问题，可视为上述Ashbaugh-Benguria猜想的推广。
英文摘要：
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.
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