Questions: Spectral Theory for Elliptic Operators

Question 1

The first eigenvalue λ₁ of -Δ on a bounded domain Ω with Dirichlet conditions satisfies:

• A. λ₁ > 0, and the corresponding eigenfunction does not change sign
• B. λ₁ = 0 with constant eigenfunction
• C. λ₁ < 0 for some domains
• D. λ₁ depends on the choice of coordinates

Answer: (A)

For Dirichlet conditions on a bounded domain, λ₁ > 0 (since ∫|∇u|²dx > 0 for any nonzero u ∈ H¹₀). The first eigenfunction φ₁ does not change sign (it is either strictly positive or strictly negative in Ω), a consequence of the Krein-Rutman theorem or the maximum principle.

Question 2

Weyl's asymptotic law states that the eigenvalue counting function N(λ) = #{λ_k ≤ λ} satisfies N(λ) ~ C_n|Ω|λ^{n/2} as λ → ∞.

• T. True
• F. False

Answer: True

Weyl's law (1911) connects the eigenvalue distribution to the volume of the domain: N(λ) ~ ω_n|Ω|/(2π)^n · λ^{n/2}, where ω_n is the volume of the unit ball in ℝⁿ. This means you can 'hear the volume' of a domain from its eigenvalues, answering part of Kac's famous question 'Can one hear the shape of a drum?'

Question 3

How is the first eigenvalue λ₁ of -Δ characterized variationally?

Model answer: λ₁ = min{∫|∇u|²dx / ∫u²dx : u ∈ H¹₀(Ω), u ≠ 0} (the Rayleigh quotient minimum)

This variational characterization (the Rayleigh-Ritz principle) gives λ₁ as the minimum of the Rayleigh quotient over H¹₀. Higher eigenvalues are characterized similarly via the min-max principle: λ_k = min over k-dimensional subspaces of the max of the Rayleigh quotient.

Question 4

The eigenfunctions of -Δ on a bounded domain with smooth boundary are smooth.

• T. True
• F. False

Answer: True

Eigenfunctions satisfy -Δφ = λφ, an elliptic equation with smooth right-hand side (λφ). By elliptic regularity, if the boundary is smooth, φ is smooth up to the boundary. On domains with corners, eigenfunctions may have limited regularity near the corners.