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Improved lower bounds for Dirichlet eigenvalues of the Laplacian and poly-Laplacian on bounded Euclidean domains
Zhengchao Ji, Yong Luo
TL;DR
Lower bounds for averaged sums of Dirichlet eigenvalues can be improved by sharpening estimates for the binary polynomials used in the Laplacian and poly-Laplacian cases. This paper derives expansions for those polynomials and uses them to obtain sharper bounds, with estimates described as best possible along these lines.
Problem
Improving eigenvalue-sum lower bounds depends on obtaining sharper estimates for the binary polynomials used in existing approaches.
Source: §1 · p. 4Method
The paper derives expansions for the binary polynomials underlying the Laplacian and poly-Laplacian lower-bound arguments.
Source: §1 · p. 5Results
Takeaways & Limitations
The polynomial estimates are described as best possible along these lines.
Source: §1 · p. 5The proof uses coordinates translated and rotated so the domain's center of mass is at the origin.
Source: §2 · p. 5
Abstract
from arXiv · showhide
In this paper, we establish Brezin-Li-Yau type lower bounds for averaged sums of Dirichlet eigenvalues of the Laplacian and poly-Laplacian on bounded domains in Euclidean spaces. By deriving expansions of two binary polynomials which may be of independent interest, we improve several existing lower bounds of this kind in the literature. Furthermore, our lower bounds are optimal in the sense that our expansions capture all positive terms, whereas previous works only provided certain lower bounds for these two binary polynomials, effectively capturing only a subset of the positive terms identified in our expansions.
1 Introduction
The introduction surveys lower bounds for Dirichlet eigenvalue sums and explains how sharper estimates for binary polynomials can improve them. The paper develops expansions for the Laplacian and poly-Laplacian cases.
1 Introduction
- Kovařík–Vugalter–Weidl's polygon bound is sharper than Melas's second term for large k [24].Their work established this type of lower bound on polygons.Source: §1 · p. 3
- The poly-Laplacian case includes the clamped plate problem when l = 2, which governs vibration of a stiff plate [20].The introduction also situates these estimates alongside work on the generalized Pólya conjecture.Source: §1 · p. 4
- Li and Yau used Fourier transformation and rearrangement arguments to bound eigenvalue sums, establishing the inequality later known as the Brezin-Li-Yau inequality.The introduction also notes that the inequality can be obtained through a Legendre transformation of Berezin's earlier result.Source: §1 · pp. 2, 4
- The key improvement strategy is to sharpen a binary-polynomial lower bound or the upper bound for M1, thereby improving eigenvalue-sum bounds.Melas used a three-term polynomial lower bound, and Ji and Xu later improved that estimate.Source: §1 · pp. 4–5
- Expansions of two binary polynomials yield a sharper Laplacian bound and a poly-Laplacian inequality that improves earlier results [20].The authors describe both polynomial estimates as best possible along their respective lines.Source: §1 · p. 5
2 Lower bounds for Dirichlet eigenvalues of the Laplacian
The section proves a Laplacian eigenvalue lower bound using a coefficient-matching polynomial lemma and Fourier-transform estimates, then states a corollary sharper than Melas’ inequality.
2.1 Proof of Theorem 2.1
- The proof estimates sums of eigenvalues through Fourier transforms of orthonormal eigenfunctions, Bessel’s inequality, and decreasing radial rearrangement.The rearrangement argument uses the coarea formula and the isoperimetric inequality.Source: §2 · pp. 6–7
2 Lower bounds for Dirichlet eigenvalues of the Laplacian
- Lemma 2.2 establishes a polynomial identity by comparing coefficients of τ^j s^(n+2−j) on both sides.The matching coefficients establish the identity used in the proof.Source: §2 · pp. 7–9
2.1 Proof of Theorem 2.1
- Corollary 2.4 gives the lower bound for bounded domains in R^n, n ≥2, and every k ≥1; the paper states it is sharper than Melas’ inequality.The proof of Theorem 2.1 precedes the corollary.Source: §2.1 · pp. 13, 15
3 Lower bounds for Dirichlet eigenvalues of the poly-Laplacian
The section establishes lower bounds for Dirichlet eigenvalues of the poly-Laplacian on bounded Euclidean domains. Theorem 3.1 improves earlier results, using polynomial identities and estimates for decreasing radial rearrangements.
3 Lower bounds for Dirichlet eigenvalues of the poly-Laplacian
- Theorem 3.1 gives a lower bound for Dirichlet eigenvalues of the poly-Laplacian on bounded domains and improves Theorem 4.1 in and Theorem 1.2 in [20].The proof uses an expansion of a binary polynomial and estimates involving decreasing radial rearrangements.Source: §3 · pp. 16–19
- The proof transforms eigenfunction information using Fourier transforms, Plancherel’s theorem, and a decreasing radial rearrangement of Q.The rearrangement is written Q*(ξ) = φ̄(|ξ|), with φ̄ assumed absolutely continuous and decreasing.Source: §3 · pp. 16–17
- The polynomial identity factors out (x − 1)^2 and yields coefficients a_j = q − 1 − j for j = 0, …, q − 2.The identity is used to establish the key equation in the proof of Theorem 3.1.Source: §3 · pp. 17–19
- Lemma 3.4 supplies an estimate for decreasing, absolutely continuous ψ under its stated conditions, which is used in proving Theorem 3.1.The proof also records the bound ∫ s^(2l+n)q(s) ds ≤ (2l+n)B, with q(s) = −ψ′(s).Source: §3 · p. 20
3.1 Proof of Theorem 3.1
- Corollary 3.5 gives a further lower bound and improves Theorem 1.2 in [20] when 2l + n ≥ 6.The proof treats cases according to whether n + 1 − l is positive.Source: §3.1 · pp. 24–26
4 Statements
The authors report no conflict of interest and no associated data.
4 Statements
- The corresponding author states that there is no conflict of interest on behalf of the authors.No conflict of interest is declared.Source: §4 · p. 26
- The manuscript has no associated data.Source: §4 · p. 26
5 Funding Declaration
The work reports support from Chinese national and Zhejiang provincial science foundations.
5 Funding Declaration
- The work was supported by the National Natural Science Foundation of China and the Natural Science Foundation of Zhejiang Province.The listed grant numbers are 12501068, 12271069, and LQN25A010001.Source: §5 · p. 26