Improved lower bounds for Dirichlet eigenvalues of the Laplacian and poly-Laplacian on bounded Euclidean domains

1. Problem Statement

The paper addresses the problem of establishing sharp lower bounds for the averaged sums of Dirichlet eigenvalues for two fundamental operators on bounded domains $\Omega \subset \mathbb{R}^n$:

1. The Laplacian ($\Delta$): Solving $-\Delta u = \lambda u$ with Dirichlet boundary conditions ($u=0$ on $\partial\Omega$).
2. The Poly-Laplacian ($(-\Delta)^l$): Solving $(-\Delta)^l \bar{u} = \Lambda \bar{u}$ with higher-order Dirichlet boundary conditions ($\bar{u} = \partial_\nu \bar{u} = \dots = \partial_\nu^{l-1} \bar{u} = 0$ on $\partial\Omega$).

The goal is to improve upon existing inequalities, specifically the Berezin-Li-Yau inequality and its subsequent refinements by Melas, Ji-Xu, and others. The authors aim to derive bounds that capture more positive terms in the asymptotic expansion, moving closer to Pólya's Conjecture (which posits that $\lambda_k \geq C_n k^{2/n}$).

2. Methodology

The authors employ a refined version of the Li-Yau method, which utilizes Fourier analysis and rearrangement inequalities. The core methodology involves:

• Fourier Transform & Plancherel's Theorem: Transforming the eigenvalue problem into the frequency domain. The sum of eigenvalues is related to the integral of $|\xi|^2$ (or $|\xi|^{2l}$) weighted by the spectral function $F(\xi) = \sum |f_j(\xi)|^2$.
• Radial Rearrangement: Replacing the spectral function $F(\xi)$ with its decreasing radial rearrangement $F^*(\xi) = \phi(|\xi|)$. This allows the problem to be reduced to estimating integrals of the form $\int s^{n-1}\phi(s) ds$ and $\int s^{n+1}\phi(s) ds$.
• Key Innovation: Binary Polynomial Expansions:
  The central technical contribution is the derivation of exact expansions for specific binary polynomials that arise in the estimation process. Previous works (e.g., Melas, Ji-Xu) provided lower bounds for these polynomials by capturing only a subset of their positive terms.
  • For the Laplacian, the authors analyze $P(s, \tau) = n s^{n+2} - (n+2)\tau^2 s^n + 2\tau^{n+2}$.
  • For the Poly-Laplacian, they analyze $P_1(s, \tau, d) = n s^{n+l} - (n+l)\tau^l s^n + l \tau^{n+l}$.
  • Instead of bounding these from below by a single term, the authors prove exact identities (Lemmas 2.2 and 3.3) that decompose these polynomials into sums of non-negative terms involving $(\tau-s)^2$.
• Optimization: By utilizing the full expansion of these polynomials, the authors derive sharper lower bounds for the integrals involving the rearranged function $\phi(s)$, leading to improved eigenvalue estimates.

3. Key Contributions

A. Theoretical Advancement in Polynomial Analysis

The paper establishes that previous lower bounds for the critical binary polynomials were suboptimal because they ignored certain positive terms in the expansion. The authors prove that:
$$n s^{n+2} - (n+2)\tau^2 s^n + 2\tau^{n+2} = \sum_{k=1}^n 2k s^{k-1}\tau^{n-k+1}(\tau-s)^2 + n(\tau-s)^2 s^n$$
This identity allows for the capture of all positive terms, making the resulting eigenvalue bounds "optimal" in the sense of this specific analytical approach.

B. Improved Lower Bounds for the Laplacian

The authors prove Theorem 2.1, which provides a new lower bound for the averaged sum of Laplacian eigenvalues:
$$\frac{1}{k}\sum_{i=1}^k \lambda_i \geq \frac{4\pi^2}{(\omega_n V)^{2/n}} k^{2/n} - \frac{S_{n+1}}{(n+1)\rho^n} \omega_n^{\frac{n-2}{n}} \alpha^{\frac{(n+2)(n-1)}{n}} k^{\frac{2-n}{n}} + c_1 \frac{n S_{n+3}}{(n+2)(n+3)\rho^{n+2}} \omega_n \alpha^{n+3} k^{-1}$$

• Significance: This improves upon the 2020 result by Ji and Xu (Eq. 1.8) and Melas's inequality (Eq. 1.4).
• Corollary 2.4: They derive a sharper version of Melas's inequality with a larger constant $\bar{c}_n$, proving $\bar{c}_n > \frac{1}{24(n+2)}$.

C. Improved Lower Bounds for the Poly-Laplacian

The authors extend their method to the poly-Laplacian, proving Theorem 3.1. This provides a sharper lower bound for the sum of eigenvalues $\Lambda_i$ of the poly-Laplacian of order $l$:
$$\frac{1}{k}\sum_{i=1}^k \Lambda_i \geq \text{Weyl Term} + \text{Correction Term}_1 - \text{Correction Term}_2$$

• Significance: This result improves upon Theorem 4.1 in [19] and Theorem 1.2 in [20] (Ji and Xu, 2025).
• Corollary 3.5: They provide a specific simplified bound involving a polynomial $Q(s)$, showing that their constant is strictly better than previous estimates (e.g., $Q(1/5) \geq 5l/2$).

4. Main Results Summary

5. Significance

• Refinement of Pólya's Conjecture: While Pólya's conjecture remains unproven for general domains, this work pushes the lower bounds closer to the conjectured asymptotic limit. The inclusion of more positive terms in the expansion reduces the "gap" between the proven lower bound and the true eigenvalue sum.
• Methodological Rigor: The paper demonstrates that the "Li-Yau method" can be pushed further by rigorous algebraic manipulation of the underlying polynomials, rather than just relying on coarse inequalities.
• Applications: These improved bounds are crucial for understanding the spectral geometry of domains, with implications for physics (e.g., vibration of plates modeled by the poly-Laplacian) and mathematical physics.
• Generalization: The results unify and improve upon a decade of literature (from 2010 to 2025) regarding eigenvalue estimates for both the Laplacian and higher-order operators.

In conclusion, Ji and Luo have successfully optimized the analytical machinery used to derive eigenvalue lower bounds, resulting in the sharpest known inequalities of the Berezin-Li-Yau type for both the Laplacian and poly-Laplacian on bounded Euclidean domains.