Math 2.0

Partial differential equations Conjecture

Pólya's conjecture on Laplacian eigenvalues

Builds on Dirichlet eigenvalueNeumann boundary conditionLaplacianLebesgue measureWeyl's law

Statement

Let d≥1d \ge 1 and let Ω⊆ℝd\Omega \subseteq \R^d be a nonempty bounded open set with Lebesgue measure ∣Ω∣\abs{\Omega}. Let ωd\omega_d be the volume of the unit ball of ℝd\R^d. Let 0<λ1≤λ2≤⋯0 < \lambda_1 \le \lambda_2 \le \cdots be the Dirichlet eigenvalues of −Δ-\Delta on Ω\Omega, where Δ\Delta is the Laplacian. If Ω\Omega is also connected with Lipschitz boundary, let 0=μ1<μ2≤μ3≤⋯0 = \mu_1 < \mu_2 \le \mu_3 \le \cdots be its eigenvalues with the Neumann boundary condition. All are repeated according to multiplicity. Let Cd=4π2ωd2/dC_d = \dfrac{4 \pi^2}{\omega_d^{2/d}}. Then, for every k≥1k \ge 1, λk≥Cd(k∣Ω∣)2/dandμk+1≤Cd(k∣Ω∣)2/d.\lambda_k \ge C_d \left( \frac{k}{\abs{\Omega}} \right)^{2/d} \qquad \text{and} \qquad \mu_{k+1} \le C_d \left( \frac{k}{\abs{\Omega}} \right)^{2/d}.

Remarks

The constant is that of Weyl's law, λk∼Cd(k/∣Ω∣)2/d\lambda_k \sim C_d (k / \abs{\Omega})^{2/d} as k→∞k \to \infty, so the conjecture says that the Weyl term is a one-sided bound for every kk, not only asymptotically. For d=2d = 2 the bounds read λk≥4πk/∣Ω∣\lambda_k \ge 4 \pi k / \abs{\Omega} and μk+1≤4πk/∣Ω∣\mu_{k+1} \le 4 \pi k / \abs{\Omega}. The conjecture is open in general. Pólya proved the Dirichlet bound for plane domains that tile the plane. Li and Yau proved the weaker bound λk≥dd+2Cd(k/∣Ω∣)2/d\lambda_k \ge \frac{d}{d + 2} C_d (k / \abs{\Omega})^{2/d} for all bounded domains, now called the Berezin–Li–Yau inequality. Filonov, Levitin, Polterovich and Sher proved both bounds for the disk. For k=1k = 1 the Dirichlet bound follows from the Faber–Krahn inequality.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptDirichlet boundary condition, concept, no evidence levelDirichlet boundary co…conceptDirichlet eigenvalue, concept, no evidence levelDirichlet eigenvalueconceptEigenvalue, concept, no evidence levelEigenvalueconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptNeumann boundary condition, concept, no evidence levelNeumann boundary cond…conceptOuter measure, concept, no evidence levelOuter measureconceptLebesgue measure, object, no evidence levelLebesgue measureobjectTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptWeyl's law, statement, no evidence levelWeyl's lawstatementPólya's conjecture on Laplacian eigenvalues, conjecture, no evidence levelPólya's conjecture on…conjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Borel sigma-algebra concept
  2. Dirichlet boundary condition concept
  3. Dirichlet eigenvalue concept
  4. Eigenvalue concept
  5. Euclidean space object
  6. Function concept
  7. Neumann boundary condition concept
  8. Outer measure concept
  9. Lebesgue measure object
  10. Total derivative concept
  11. Function of several variables concept
  12. Vector-valued function concept
  13. Differentiable function of several variables concept
  14. Partial derivative concept
  15. Vector field concept
  16. Gradient concept
  17. Divergence concept
  18. Laplacian concept
  19. Weyl's law statement
  20. Pólya's conjecture on Laplacian eigenvalues conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Weyl_law
About this page
TagF68QFEYFY0
URIm2://F68QFEYFY0
Kindconjecture
WritingW0 stub
MSC35P15, 49R05, 35P20
StatusThis copy of the site has no build data. Run m2 status F68QFEYFY0.
AboutDirichlet eigenvalue
UsesDirichlet eigenvalue, Neumann boundary condition, Laplacian, Lebesgue measure, Weyl's law

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