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Improved Berezin-Li-Yau inequality and Kr\"oger inequality and consequences
Zaihui Gan, Renjin Jiang, Fanghua Lin
TL;DR
The paper addresses quantitative strengthening of Berezin–Li–Yau and Kröger inequalities, including Weidl’s open question about a correction term. Using moment-of-inertia estimates, it obtains quantitative improvements and shows that infinitely many Dirichlet eigenvalues satisfy Pólya’s conjecture for n ≥ 3, and infinitely many Neumann eigenvalues do so for n ≥ 5 when the Neumann spectrum is discrete.
Problem
Weidl raised the open question of whether Kröger’s inequality can be strengthened by including a correction term.
Source: §1 · p. 5Method
The paper uses the moment-of-inertia method to estimate Dirichlet Riesz means and gives a quantitative improvement to Kröger’s inequality.
Source: §1 · pp. 4–5Results
For bounded open domains, infinitely many Dirichlet eigenvalues satisfy Pólya’s conjecture when n ≥ 3, and infinitely many Neumann eigenvalues do so when n ≥ 5 if the Neumann spectrum is discrete.
Source: §1 · p. 5Takeaways & Limitations
The quantitative Kröger improvement answers Weidl’s open question about adding a correction term.
Source: §1 · p. 5For arbitrary domains without boundary regularity, the boundary measure may be infinite, so Weyl’s sharper asymptotic formula need not hold; the Neumann result also assumes a discrete spectrum.
Source: §1 · p. 5, §3 · p. 20
Abstract
from arXiv · showhide
We provide quantitative improvements to the Berezin-Li-Yau inequality and the Kr\"oger inequality, in $\mathbb{R}^n$, $n\ge 2$. The improvement on Kr\"oger's inequality resolves an open question raised by Weidl from 2006. The improvements allow us to show that, for any open bounded domains, there are infinite many Dirichlet eigenvalues satisfying P\'olya's conjecture if $n\ge 3$, and infinite many Neumann eigenvalues satisfying P\'olya's conjecture if $n\ge 5$ and the Neumann spectrum is discrete.
1 Introduction
The introduction presents quantitative improvements to the Berezin-Li-Yau and Kröger inequalities, then uses them to obtain new Pólya-conjecture results for Dirichlet and Neumann eigenvalues. It situates these results against the conjecture’s limited prior scope and notes the assumptions and boundary-regularity limits relevant to their reach.
1 Introduction
- Pólya’s conjecture remained open for general domains, with prior results covering tiling, product, and certain other domains or eigenvalues.The introduction also notes results for balls, annuli, Lipschitz domains, and some initial eigenvalues on general domains.Source: §1 · p. 2
- For the Dirichlet Riesz means, the authors adopt Melas’ moment-of-inertia method to obtain estimates.Source: §1 · p. 4
- Theorem 1.1 gives an improved Berezin-Li-Yau estimate whose constant C1(n) improves on Melas’ earlier improvement.The factor involving the Gamma function in C1(n) has a uniformly positive lower bound of about 1.014 for n ≥ 3.Source: §1 · p. 4
- Without boundary regularity, the boundary measure may be infinite, so the sharper Weyl asymptotic formula cannot hold in such general settings.Source: §1 · p. 5
- For bounded open domains, infinitely many Dirichlet eigenvalues satisfy Pólya’s conjecture when n ≥ 3.The authors derive this qualitative result from a more quantitative theorem that guarantees at least one such eigenvalue in an interval of a certain length.Source: §1 · p. 5
- The paper quantitatively improves Kröger’s inequality, addressing Weidl’s open question about adding a correction term.Earlier improvements had not provided explicit constants.Source: §1 · p. 5
- For bounded open domains with discrete Neumann spectrum, infinitely many Neumann eigenvalues satisfy Pólya’s conjecture when n ≥ 5.Source: §1 · p. 5
- The authors describe the Neumann improvement as weaker because the Fourier transform does not turn the Neumann Laplacian into polynomial multipliers.For Dirichlet eigenvalues, the corresponding sum is an integral over R^n.Source: §1 · p. 6
2 Improved Berezin-Li-Yau inequality
The section proves a sharper Berezin-Li-Yau estimate and derives improved eigenvalue estimates. Its refinement yields a uniformly positive factor in dimensions n ≥ 3 and supports a qualitative conclusion in Theorem 2.8.
Theorem 2.4
- Theorem 2.4 gives an improvement over the original Berezin-Li-Yau inequality.The proof uses a realization function supported on an interval, with a linear profile there and zero elsewhere.Source: §2 · pp. 12, 14
Theorem 1.1 and Remark 2.6
- The cited analysis states that a quantity admits a uniformly positive lower bound for every n ≥ 3, but the supplied passages do not identify the quantity or specify the bound.Source: §2 · p. 18
Corollary 2.7
- Theorem 1.1 also yields an eigenvalue estimate with an improved constant C1(n).The estimate is stated for bounded open sets in dimensions n ≥ 2.Source: §2 · p. 18
Theorem 2.8
- Theorem 2.8 supplies the qualitative conclusion used to establish the stated asymptotic result for Dirichlet eigenvalues.The supplied proof passage says the conclusion follows from Theorem 2.8 as the index tends to infinity.Source: §2 · pp. 18, 20
3 Improved Kr¨oger inequality
This section proves an improved Kröger inequality under a discrete-spectrum assumption for the Neumann Laplacian. The resulting estimate supports a qualitative conclusion for Neumann eigenvalues in the stated dimensional range.
Theorem 3.1
- Theorem 3.1 considers a bounded open set Ω in R^n, for n ≥ 3, and assumes that the Neumann Laplacian −∆N has discrete spectrum.Source: §3 · p. 23
Proof of Theorem 1.3
- The proof obtains an interval on which the Neumann counting expression FN(γ) is increasing.The supplied passage connects this increase to positivity of its derivative on that interval.Source: §3 · p. 24
Consequence of Theorem 3.1
- The section’s concluding argument derives its qualitative Neumann eigenvalue result from Theorem 3.1 as the index tends to infinity.The conclusion is stated for the discrete-spectrum setting.Source: §3 · pp. 20, 23, 25