PaperlayerView exact revision on arXiv

Source-linked AI summary

Lower bounds for the spectrum of the Laplace and Stokes operators

Alexei A. Ilyin

arXiv:0909.2818v1math.AP

TL;DR

The paper addresses lower bounds for sums of Laplace and Stokes eigenvalues, including bounds relevant to Navier–Stokes attractor theory. It extends an existing approach and solves the associated minimization problem exactly. The resulting asymptotic bounds have a positive linear term whose coefficient is n(n + 2)/2 times that in the earlier estimate.

  • Problem

    The paper studies lower bounds for sums of eigenvalues of the Laplace and Stokes operators, which are relevant in the theory of Navier–Stokes attractors.

    Source: §1 · pp. 1–2

  • Method

    The authors extend the approach of to the Stokes operator and solve the corresponding minimization problem exactly.

    Source: §1 · p. 2

  • Results

    The asymptotic second-term coefficient is n(n + 2)/2 times that in the earlier estimate (1.2), and the second term is positive and linear in m.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The resulting lower bounds apply to sums of eigenvalues of both the Laplace and Stokes operators.

    Source: §1 · p. 4

    The stated setting is an open subset of R^n with finite n-dimensional Lebesgue measure.

    Source: §2 · p. 4

Abstract

from arXiv · show

We prove Berezin--Li--Yau-type lower bounds with additional term for the eigenvalues of the Stokes operator and improve the previously known estimates for the Laplace operator. Generalizations to higher-order operators are given.

1. Introduction

The paper extends Berezin–Li–Yau-type lower bounds to the Stokes operator and improves Laplace bounds by solving the associated minimization problem exactly. Its asymptotic bounds include a larger positive linear term, with further explicit results in dimensions 2, 3, and 4 and an extension to the Dirichlet bi-Laplacian.

1. Introduction

  • Section 3 finds the exact solution of minimization problem (1.8); for large m, the coefficient of m in the second term on the right-hand side of (1.11) is n(n + 2)/2 times that in (1.2).Source: §1 · p. 3
  • The asymptotic expansion has a positive linear second term for every n, with coefficient n(n + 2)/2 times that in (1.9), while its third term is negative.The expansion proceeds in descending powers m^(1+2/n), m, m^(1−2/n), m^(1−4/n), ….Source: §1 · p. 3
  • For large m, dropping the negative third term yields lower bounds whose linear-term coefficient is n(n + 2)/2 times that in the earlier estimate (1.2).The bounds apply to sums of eigenvalues of both the Laplace and Stokes operators, with a nonnegative remainder ε_n(m) = O(m^(−2/n)).Source: §1 · pp. 3–4
  • In dimensions 2, 3, and 4, the paper derives explicit formulas for the minimization solution; in dimension 2, the exact formula coincides with the first three asymptotic terms.For dimensions 3 and 4, the authors use explicit formulas and choose β values close to 1 to compensate for the negative third-term contribution.Source: §1 · p. 4
  • The paper also proves two-term lower bounds for the Dirichlet bi-Laplacian.It notes that other constant-coefficient elliptic equations and systems with Dirichlet conditions can be treated similarly, but restricts its treatment to the bi-Laplacian.Source: §1 · p. 4

2. Estimates for orthonormal vector functions

The section develops Fourier-transform estimates for orthonormal vector functions and applies them to Stokes eigenvalue bounds. The divergence-free condition sharpens the vector estimate by replacing a factor n with n−1.

2. Estimates for orthonormal vector functions

  • The Stokes operator is defined through the gradient inner product on V and has a discrete positive spectrum of eigenvalues λk.The domain Ω is an open subset of R^n with finite measure.Source: §2 · pp. 4–5
  • Suborthonormal families can arise by projecting an orthonormal family, and the section extends Fourier-transform estimates to these families.The projected families ηk = Pϕk and ξk = (I − P)ϕk are both suborthonormal.Source: §2 · p. 5
  • For divergence-free orthonormal vector functions, the estimate’s factor n is replaced by n−1, a refinement essential to the Stokes Li–Yau bounds.The section also establishes estimates for Fourier transforms of (sub)orthonormal families.Source: §2 · pp. 5–7

3. Minimization problem

The eigenvalue-bound problem is reduced to a one-dimensional minimization over decreasing functions. Its exact minimizer yields explicit bounds and an asymptotic expansion that strengthens the Laplace estimate’s second term.

3. Minimization problem

  • The minimization reduces to a one-dimensional problem for a decreasing, absolutely continuous function.The Stokes and Laplace cases differ through the parameters M and L in the minimization problem.Source: §3 · pp. 7–8
  • The minimizing profile is Φs(r): it stays at M through s, decreases linearly to zero by s + M/L, and gives the exact solution of the minimization problem.The solution is expressed using the unique positive root t(m*) of equation (3.6).Source: §3 · pp. 8–9
  • Substituting the Laplace and Stokes parameters into the minimization solution yields the lower bounds for sums of their first m eigenvalues.The proof takes the first normalized eigenvectors of the Stokes problem, with the Laplace proof proceeding similarly.Source: §3 · pp. 7, 10

The case n = 2.

For n = 2, the minimization solution has an explicit formula that coincides with its first three asymptotic terms. The resulting theorem gives lower bounds for the Laplace and Stokes eigenvalues.

The case n = 2.

  • The two-dimensional case is presented as the simplest case, with the most complete results..Source: The case n = 2 · p. 10
  • For n = 2, the positive root t(m*) of (t + 1)^3 − t^3 = m* is used to calculate the last factor, and ΣM,L(m) equals Σ0(m) for n = 2.Source: The case n = 2 · p. 10
  • Theorem 4.1 states lower bounds for eigenvalues of both the Laplace and Stokes operators when n = 2.The proof substitutes the corresponding M and L parameters for each operator.Source: The case n = 2 · pp. 10–11

The case n = 4.

For n = 4, the paper gives explicit minimization formulas and compares them with the asymptotic expression. The reported coefficients are close to one for both operators.

The case n = 4.

  • For the four-dimensional case, the positive root t(m*) of equation (3.6) with n = 4 is described by a biquadratic in η = t + 1/2.Source: The case n = 4 · p. 11
  • For n = 4, the reported coefficients are βL = 0.983 and βS = 0.978 for the Laplace and Stokes operators, respectively.The more precise values given are βL = 0.9835… and βS = 0.9786….Source: The case n = 4 · p. 11
  • For n = 4, the exact solution ΣM,L(m) is greater than the three-term asymptotic expression Σ0(m).The proof notes that the three terms on the right match those in the n = 4 asymptotic expansion.Source: The case n = 4 · p. 11

The case n = 3.

For n = 3, the Laplace and Stokes bounds use distinct coefficients, and the sum bound satisfies ΣM,L(m) > Σ0(m)n=3. The cubic root needed in the calculation is given by Cardano’s formula.

The case n = 3.

5. Further examples. Dirichlet bi-Laplacian

The section extends the minimization approach to the Dirichlet bi-Laplacian, presenting an eigenvalue bound for n = 2. The leading coefficient of its m3 term is sharp.

5. Further examples. Dirichlet bi-Laplacian

  • The authors note that other constant-coefficient elliptic equations and systems with Dirichlet boundary conditions can be treated similarly.Source: §5 · p. 13
  • The Dirichlet bi-Laplacian bound is obtained from a minimization problem using the same solution approach as Proposition 3.1.Source: §5 · p. 13
  • The discussion restricts the worked example to the least technical, two-dimensional case.Source: §5 · p. 13
  • For n = 2, Theorem 5.1 gives an eigenvalue bound for the Dirichlet bi-Laplacian.Source: §5 · p. 14
  • The coefficient of the leading m3 term in (5.6) is sharp.Source: §5 · p. 14
Loading 0909.2818v1…