The role of critical eigenvalues and eigenfunctions in a class of singularly perturbed problems

Author

Vagelis Stefanopoulos

Date of Award

12-1993

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

N. D. Alikakos

Committee Members

V. Alexiades, D. Hinton, P. Schaefer, H. Simpson, S. Georghiou

Abstract

We consider the eigenvalue problem -ε²Δ𝜓 + f’(u_ε)𝜓i = µ𝜓 , with Neumann boundary conditions, in a smooth bounded domain Ω in ℝ² , arising from the evolution problem u_t = ε²Δu - f(u) with the same boundary conditions. Here f is the derivative of a double well potential with two equal minima at ;plusmn;1 and u_εis a smooth function with a sharp interface γ contained in Ω.

The goal is to characterize the eigenvalues μ that vanish in the limit ε → 0 (critical), and their corresponding eigenfunctions. It is shown that these eigenvalues are related to the eigenvalues λ of a regular Sturm-Liouville problem on the curve γ by μ = ε²λ + Ο(ε³) . By changing to local coordinates r(= signed distance from γ) and s(=arclength on γ) in a neighborhood of γ, and then stretching variables ( η = r/ε) Alikakos and Fusco have shown that the critical eigenfunctions, appropriately normalized, separate in the limit ε → 0 to U'(η)Θ(s), where U' is the derivative of the solution of U’’- f(U) = 0, U(0) = 0 , U(± ∞) = ±1 and Θ(s) is a periodic function. Elaborating on this separation result, we prove that Θ is an eigenfunction of the above mentioned Sturm-Liouville problem. We thus establish the following characterization for the eigenpair (μ, ψ): μ_i = ε²λ_i + O(ε³) , ψ_i = U'(r/ε)Θ_i(s) + Ο(ε), i = 1, 2,...,m, where mis fixed but arbitrary. The proof involves among other things spectral analysis of radial operators, justification of formal asymptotic expansions, and perturbation theory.

The same analysis carried to the case where u_ε is a radial equilibrium of the corresponding evolution problem with an interface γ, shows that every such solution is unstable.

Recommended Citation

Stefanopoulos, Vagelis, "The role of critical eigenvalues and eigenfunctions in a class of singularly perturbed problems. " PhD diss., University of Tennessee, 1993.
https://trace.tennessee.edu/utk_graddiss/10781