[Paper Review] Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity

Michel Chipot, Prosenjit Roy+1 more|arXiv (Cornell University)|Mar 1, 2014

Advanced Mathematical Modeling in Engineering9 references3 citations

TL;DR

This paper investigates the asymptotic behavior of eigenvalues and eigenfunctions for elliptic operators with mixed Neumann-Dirichlet boundary conditions on long cylindrical domains as the length ℓ → ∞. It establishes that the first eigenvalue converges to min(ν₊, ν₋), the first eigenvalues on semi-infinite cylinders, which is strictly less than μ¹ (the first eigenvalue on the cross-section ω) when A₁₂·∇W₁ ≢ 0, revealing a 'gap phenomenon' absent in Dirichlet cases.

ABSTRACT

We analyze the asymptotic behavior of eigenvalues and eigenfunctions of an elliptic operator with mixed boundary conditions on cylindrical domains when the length of the cylinder goes to infinity. We identify the correct limiting problem and show in particular, that in general the limiting behavior is very different from the one for the Dirichlet boundary conditions.

Motivation & Objective

• To analyze the asymptotic behavior of eigenvalues and eigenfunctions of elliptic operators with mixed boundary conditions on cylindrical domains as the length ℓ → ∞.
• To identify the correct limiting problem for the first eigenvalue in the case of mixed Neumann-Dirichlet conditions, contrasting it with the Dirichlet case studied previously.
• To characterize the limiting behavior through eigenvalue problems on semi-infinite cylinders Ω₊∞ and Ω₋∞, particularly identifying limₗ→∞ λₗ¹ = min(ν₊, ν₋).
• To explain the 'gap phenomenon'—where the limit is strictly less than μ¹—by analyzing boundary layer effects near the ends Γₗ.
• To generalize the results to domains expanding in multiple directions, showing that the gap persists under a non-degeneracy condition on A₁₂·∇W₁.

Proposed method

• Formal analysis of the eigenvalue problem −div(A(X₂)∇u) = σu on Ωₗ = (−ℓ,ℓ)×ω with Dirichlet on lateral boundary γₗ and Neumann on end faces Γₗ.
• Use of variational methods and Rayleigh quotients to derive upper bounds on λₗ¹, leading to limsup estimates via test functions on reduced domains.
• Introduction of auxiliary eigenvalue problems on semi-infinite cylinders Ω₊∞ = (0,∞)×ω and Ω₋∞ = (−∞,0)×ω with zero Dirichlet conditions on lateral parts.
• Derivation of the key identity: limₗ→∞ λₗ¹ = min(ν₊, ν₋), where ν₊, ν₋ are the first eigenvalues on the semi-infinite cylinders.
• Application of a generalized min-max principle and decomposition of the energy functional using block structure of A(X₂) = [A₁₁ A₁₂; A₁₂ᵀ A₂₂].
• Use of the identity min_{Z₁} (Bz)·z = (B₂₂Z₂)·Z₂ − (B₁₁⁻¹B₁₂Z₂)·B₁₂Z₂ to decouple the energy and derive a lower bound involving A₂₂ and A₁₁⁻¹A₁₂.

Experimental results

Research questions

• RQ1 What is the limiting behavior of the first eigenvalue λₗ¹ as the length ℓ → ∞ for elliptic operators with mixed Neumann-Dirichlet boundary conditions?
• RQ2 How does the limiting eigenvalue compare to μ¹, the first eigenvalue on the cross-section ω, and under what conditions is it strictly smaller?
• RQ3 What role do boundary layer effects near the ends Γₗ play in determining the asymptotic eigenvalue?
• RQ4 Can the limit limₗ→∞ λₗ¹ be characterized via eigenvalue problems on semi-infinite cylinders Ω₊∞ and Ω₋∞?
• RQ5 How does the structure of the coefficient matrix A(X₂), particularly the off-diagonal block A₁₂, influence the asymptotic eigenvalue?

Key findings

• The limit limₗ→∞ λₗ¹ exists and equals min(ν₊, ν₋), where ν₊ and ν₋ are the first eigenvalues of the operator −div(A(X₂)∇u) on the semi-infinite cylinders Ω₊∞ and Ω₋∞ with Dirichlet conditions on the lateral boundary.
• When A₁₂·∇W₁ ≢ 0 a.e. on ω, the limit limₗ→∞ λₗ¹ is strictly less than μ¹, the first eigenvalue on the cross-section ω, demonstrating a 'gap phenomenon' not present in Dirichlet problems.
• If A₁₂·∇W₁ ≡ 0 a.e. on ω, then λₗ¹ = μ¹ for all ℓ > 0, indicating no gap and full convergence to the cross-sectional eigenvalue.
• The result is generalized to domains expanding in p > 1 directions: limsupₗ→∞ λₗ¹ < μ¹ if A₁₂·∇W₁ ≢ 0 ∈ ℝᵖ, with the same limiting behavior via semi-infinite cylinder eigenvalues.
• The lower bound for λₗ¹ is derived via a variational decomposition: ∫(A∇u)·∇u ≥ ∫[A₂₂∇u·∇u − (A₁₁⁻¹A₁₂∇u)·A₁₂∇u], leading to a new eigenvalue problem with operator −div(A₂₂∇w) + div(A₁₂ᵀA₁₁⁻¹A₁₂∇w).
• The infimum Λ¹ in the variational characterization (7.7) equals μ¹ if A₁₂·∇W₁ ≢ 0, confirming that the lower bound matches the known upper bound and thus λₗ¹ → min(ν₊, ν₋) < μ¹.