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Universal lower bounds for Dirichlet eigenvalues
Stefan Steinerberger
TL;DR
Universal lower bounds for Dirichlet eigenvalues are available, but Pólya's one-sided Weyl-law conjecture remains difficult. The paper exploits flexibility in the Li–Yau approach to obtain improved inequalities, including two-point bounds that are strictly stronger than Berezin–Li–Yau alone.
Problem
Pólya's one-sided Weyl-law conjecture for Dirichlet eigenvalues remains difficult, while Berezin–Li–Yau estimates match Weyl's law only up to the factor d/(d+2).
Source: §1.1 · pp. 1–2Method
The paper exploits rigidity in the Li–Yau proof and reformulates its estimate using a parameter measuring how much mass lies inside the optimal ball.
Source: §1.2 · p. 2Results
The paper establishes two-point inequalities that are strictly stronger than Berezin–Li–Yau alone, with a nontrivial improvement that diminishes as ℓ grows relative to n.
Source: §1.3 · pp. 2–3Takeaways & Limitations
In d = 2, the correction term can asymptotically reach half the leading-order term.
Source: §1.2 · p. 2The averaging estimate's constant is suboptimal; it cannot exceed 2/(d+2), and the proof does not track it optimally.
Source: §1.4 · p. 3
Abstract
from arXiv · showhide
Let $\Omega \subset \mathbb{R}^d$ be a bounded domain and let $\lambda_1, \lambda_2, \dots$ denote the sequence of eigenvalues of the Laplacian subject to Dirichlet boundary conditions. We consider inequalities for $\lambda_n$ that are independent of the domain $\Omega$. A well--known such inequality follows from the Berezin--Li--Yau approach. The purpose of this paper is to point out a certain degree of flexibility in the Li--Yau approach. We use it to prove a new type of two-point inequality which are strictly stronger than what is implied by Berezin-Li-Yau itself. For example, when $d=2$, one has $ 2 \lambda_n + \lambda_{2n} \geq 10 \pi n/|\Omega|.$
1. Introduction and Results
Berezin–Li–Yau estimates provide universal lower bounds for Dirichlet eigenvalues, while the related one-sided Weyl-law conjecture remains difficult. The paper exploits flexibility in the Li–Yau approach to prove improved universal inequalities, including two-point bounds and an averaging estimate.
1.1. Introduction.
- Pólya's one-sided Weyl-law conjecture remains difficult despite Berezin–Li–Yau estimates that match Weyl's law up to the factor d/(d+2).The conjecture was verified for some special domains and recently for the Euclidean ball.Source: §1.1 · pp. 1–2
1.2. Main Result.
- The rigidity of the Li–Yau proof motivates a simple, unconditional universal improvement.For d = 2, the correction can be asymptotically as large as half the leading-order term.Source: §1.2 · p. 2
1.3. Two-Point Inequalities.
- The two-point inequalities are strictly stronger than Berezin–Li–Yau alone when ℓ is sufficiently large, though their improvement diminishes as ℓ grows relative to n.For d = 2, the condition requires ℓ ≥ 0.618n.Source: §1.3 · pp. 2–3
1.4. An Averaging Estimate.
- Theorem 3 gives universal inequalities showing that Berezin–Li–Yau cannot be nearly sharp on average; its constant is not optimized.The constant cd cannot exceed 2/(d+2), and would attain that value if Pólya's conjecture were true.Source: §1.4 · p. 3
2. Motivation and Proof of Theorem 1
The section reviews the Li–Yau Fourier-transform argument and develops a mass-concentration version with a parameter η. This flexibility motivates an improvement when the Li–Yau bound is nearly sharp.
2.1. Summary of Li–Yau.
- Li–Yau minimizes an integral of ∥ξ∥^2g(ξ) with fixed L1 mass by concentrating that mass in a disk, subject to g(ξ) ≤ 1.This yields the desired bound.Source: §2.1 · p. 4
2.2. A counterfactual proposition.
- If Li–Yau were sharp for λ_n, its proof would nearly imply Pólya’s conjecture for λ_{n+1}, though exact sharpness is impossible.The counterfactual motivates adapting the proof to the almost-sharp case.Source: §2.2 · p. 4
2.3. A Mass Concentration Version.
- The parametrized Li–Yau estimate introduces η to measure mass concentration in the optimal ball; η=0 recovers Li–Yau, while near-sharpness requires η near zero.The lemma asserts such a parameter exists for the eigenfunctions u_1, …, u_k.Source: §2.3 · p. 5
- Applying the bathtub principle twice bounds the inside mass using a reduced-radius ball and places the remaining mass in a shell just outside the original ball.The shell construction uses that the integrand is radial and monotonically increasing.Source: §2.3 · pp. 5–6
2.4. Proof of Theorem 1.
- Near-sharpness at λ_k would localize subsequent Fourier mass outside the corresponding ball, allowing a second bathtub-principle step to improve the bound.The final inequality remains valid at k=n, but adds nothing new there.Source: §2.4 · p. 7
3. Proof of Theorem 2
The proof minimizes an expression that is linear in η by checking the coefficient of η and, when it is nonnegative, setting η to zero.
4. Proof of Theorem 3
The proof handles n=1 explicitly and uses that case to obtain a weaker form of Theorem 3 with a constant decaying in n. It then treats small and sufficiently large n separately.
- For n=1, the authors establish an associated gap, which yields a weaker form of Theorem 3 with a constant decaying in n.Source: §4 · p. 8
- The ratio in the n=1 comparison is minimized at d=2, where it is approximately 2.89.The passage then states that an n=1 bound follows.Source: §4 · p. 8
- Lemma 1 permits assuming n is sufficiently large; the proof instead uses Lemma 2 for small n and the large-n argument otherwise.The large-n argument is for n sufficiently large depending only on d.Source: §4 · p. 9
- The concluding argument obtains ε_1>0 depending only on d and chooses the final constant as ε_1/8.Source: §4 · p. 10