Two dimensional Berezin-Li-Yau inequalities with a correction term

Hynek Kovarik; Semjon Vugalter; Timo Weidl

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Two dimensional Berezin-Li-Yau inequalities with a correction term

Hynek Kovarik, Semjon Vugalter, Timo Weidl

arXiv:0802.2792v1Feb 20, 2008math.SP

TL;DR

Melas's correction lacks the k-order predicted by the second Weyl term and does not explicitly reflect boundary effects. The paper refines condition (7) to add a positive correction tied to that asymptotic term; for δ<1, the bound improves Melas's result.

Problem
Melas's additional term does not have the k-order predicted by the second term in the asymptotics, and its coefficient does not explicitly reflect the domain boundary.
Source: §1 · p. 2
Method
The paper improves condition (7), rather than relying on regularity of F*, to add a positive correction reflecting the second term in the asymptotics.
Source: §1 · p. 2, §2.3 · p. 4
Results
For δ<1, the estimate improves the Melas bound, while the asymptotic formula implies δ≥1/2.
Source: §2.3 · p. 5
Takeaways & Limitations
The new second-term coefficient reflects the expected effect of the domain boundary.
Source: §3.3 · p. 7
The boundary term is visible only for sufficiently large k, and the condition k≥k_j cannot be removed.
Source: §3.3 · p. 7

Abstract

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We improve the Berezin-Li-Yau inequality in dimension two by adding a positive correction term to its right-hand side. It is also shown that the asymptotical behaviour of the correction term is almost optimal. This improves a previous result by Melas.

1 Introduction

In dimension two, the paper seeks to improve the Li-Yau inequality by adding a positive correction that reflects the second asymptotic term. The motivation is that the leading constant is sharp, while the existing Melas correction does not capture the predicted order or boundary dependence.

1 Introduction

The leading constant in the Li-Yau bound cannot be improved, but its positive second asymptotic term motivates adding a lower-order term.The second asymptotic term reflects the effect of the domain boundary.Source: §1 · p. 1
Melas added a dimension-dependent correction, but it has neither the order predicted by the second asymptotic term nor explicit boundary dependence.The second asymptotic coefficient reflects the boundary, whereas the coefficient V/I in the Melas bound does not.Source: §1 · p. 2
The authors aim to improve the Li-Yau and Melas bounds with a positive contribution reflecting the second asymptotic term.The presentation is restricted to dimension two.Source: §1 · p. 2
The prior approach in [13] applies only for σ ≥ 3/2, so it cannot be used for the paper’s estimates at σ = 1.The authors therefore develop a different approach.Source: §1 · p. 2

2 Preliminaries

The section revisits the Li–Yau and Melas bounds, then identifies a boundary-sensitive source for a positive correction to the two-dimensional inequality. The correction improves Melas’ bound for δ < 1, while the asymptotic formula requires δ ≥ 1/2.

2 Preliminaries

The preliminaries define Θ as the Heaviside function, N_t as the count of Dirichlet-Laplacian eigenvalues at most t, and [s] as the integer part of s.Source: §2 · p. 2

2.1 Li-Yau bound revisited

For d = 2, extending the eigenfunctions by zero and applying Parseval and radial rearrangement leads to the Li–Yau bound via a minimizing function.The minimizer is specified by a radius r_k, and substituting it yields inequality (1).Source: §2.1 · pp. 2–3

2.2 Melas’ improvement revisited

Melas’ bound adds a regularity condition on F* and uses its resulting minimizer to obtain an improved lower bound.The minimizer is piecewise linear over radial intervals; a further estimate gives a uniform bound.Source: §2.2 · pp. 3–4

2.3 The new correction term

The improvement comes from refining condition (7), not from further regularity of F*; the tail of the Fourier coefficients reflects the boundary of Ω.The tail cannot decay arbitrarily fast because the Fourier series approximates a function discontinuous on ∂Ω, yielding a sharper upper bound on F and an additional inequality term.Source: §2.3 · pp. 4–5
For δ < 1, the correction improves Melas’ bound; the asymptotic formula requires δ ≥ 1/2.The correction has a positive coefficient A(ε, δ).Source: §2.3 · p. 5

3 Main results

Section 3 states improved lower bounds for polygons and, under a boundary-regularity assumption, general planar domains. The correction reflects boundary geometry, becomes visible only for sufficiently large k, and can be optimized through the choice of boundary segments.

3.1 Case 1: Polygons

The polygon setup measures each side’s separation from the rest of the boundary using the distance from its middle third.For side p_j, d_j is the distance from its middle third to ∂Ω \ p_j.Source: §3.1 · p. 5
Theorem 1 gives a lower bound for a polygon with n sides, for any k ∈ N and α ∈ [0, 1].The result is stated for polygons and uses the lengths of their sides.Source: §3.1 · p. 5

3.2 Case 2: General domains

For general domains, the boundary portions are divided into equal-length pieces using two tilings determined by curvature and length.The construction defines distances from selected boundary pieces to the remainder of the boundary.Source: §3.2 · p. 6
Theorem 2 extends the lower bound to domains satisfying Assumption A, which requires C2-smooth boundary parts.The boundary portions Γ_j are the specified smooth parts of ∂Ω.Source: §3.2 · p. 6

3.3 Remarks

The correction’s boundary contribution is visible only when k is sufficiently large; folding the boundary can increase total length while leaving the first k eigenvalues almost unchanged.This is why the condition k ≥ k_j cannot be removed and the second term cannot simply be proportional to total boundary length.Source: §3.3 · p. 7
Approximating a general domain by polygons can erase the second-term contribution, while choosing boundary segments appropriately can optimize the correction for the domain’s geometry.For an open ball, the coefficients k_j grow without bound along polygonal approximations; ε(k) tends to zero, although k_j may be large.Source: §3.3 · p. 7

4 Proof for polygons

The polygon proof turns the correction-term estimate into a lower bound on contributions from boundary-adjacent squares. It combines eigenfunction estimates and square-level integral bounds, then sums the contributions to obtain Theorem 1.

4 Proof for polygons

The proof targets integrals over boundary-adjacent squares of side length of order λ^-1/2, using upper bounds on eigenfunction normal derivatives at ∂Ω.These square contributions are estimated from below because the comparison function vanishes on ∂Ω.Source: §4 · p. 7

4.1 Eigenfunctions estimates

The eigenfunction estimates near ∂Ω and the ensuing corollary provide a technical input to the proofs of Theorems 1 and 2.The section develops L2 estimates on derivatives of eigenfunctions in the vicinity of the boundary.Source: §4.1 · p. 8, §4.2 · p. 10

4.2 Lower bound on a square

An integral estimate on a square is established as a central ingredient in the proofs of Theorems 1 and 2.Proposition 2 applies to a complex-valued function satisfying a boundary-vanishing condition.Source: §4.2 · p. 12

4.3 Proof of Theorem 1

Summing the lower bounds over non-overlapping boundary squares yields the estimate used to prove (21), Theorem 1’s inequality.The squares are kept inside Ω and do not overlap.Source: §4.3 · pp. 13–14

5 Proof for general domains

For general domains, the proof straightens selected boundary pieces by extending the domain, then applies the polygon strategy. The extension is designed to keep its added volume controlled, and the resulting estimates yield Theorem 2.

5 Proof for general domains

The general-domain strategy extends Ω with small bumps to create straight boundary pieces, then uses eigenvalue monotonicity to transfer lower bounds back to Ω.The extension’s volume must remain close to V so it does not diminish the correction term’s effect on the leading term.Source: §5 · p. 14

5.1 Step 1: Some geometrical remarks

When the curvature and arc-length condition κ0|s′ − s′′| ≤ π/4 holds, the boundary arc between A and B can be represented as a graph in local coordinates.The graph has slope bounded in absolute value by 1.Source: §5.1 · p. 15

5.2 Step 2: Approximation of the boundary

The boundary-approximation step selects disjoint arcs along Γj and connects their endpoints by straight segments in local coordinates.The arcs are chosen on portions of the boundary between partition points.Source: §5.2 · p. 16

5.3 Step 3: Extended domain Ωe.

The construction replaces selected curved boundary pieces with straight segments and smooth connections to form Ωe.The added area is controlled during construction, and Corollary 3 provides estimates on the extended domain’s volume.Source: §5.3 · pp. 17–18

5.4 Proof of Theorem 2

Finally, inequalities (62) and (16) are combined to obtain (26); the preceding argument uses the Li-Yau type minimiser (10), replacing V/4π2 by Me(p, λ).Source: §5.4 · pp. 19–20