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arxiv 2403.08070 v3 pith:VZUPFUJX submitted 2024-03-12 math.AP math.DG

On the Ashbaugh-Benguria type conjecture about lower-order Neumann eigenvalues of the Witten-Laplacian

Ruifeng Chen , Jing Mao This is my paper
classification math.AP math.DG
keywords analconjectureeigenvaluesneumannashbaugh-benguriaconclusioninequalitymath
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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 Szeg\H{o}-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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Cited by 2 Pith papers

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  1. A proof of the Ashbaugh--Benguria conjecture for reciprocal sums of Neumann eigenvalues

    math.SP 2026-06 unverdicted novelty 9.0

    Proves that the ball minimizes the sum of reciprocals of the first m nonzero Neumann eigenvalues among smooth bounded domains of fixed volume in R^m, with equality only for balls.

  2. The Ashbaugh--Benguria reciprocal-gap conjecture for Dirichlet eigenvalues

    math.AP 2026-07 unverdicted novelty 8.0

    Proves that for bounded domains in R^N (N≥2), the sum from i=1 to N of λ1/(λ_{i+1}-λ1) is at least N/(j_{N/2,1}^2/j_{N/2-1,1}^2 -1), with equality precisely when the domain is a ball up to H^1-capacity zero.