Singularly perturbed spectral problems with Neumann boundary conditions

TL;DR

This paper investigates the asymptotic behavior of the principal eigenvalue and eigenfunction for a singularly perturbed Neumann spectral problem, using Hamilton-Jacobi equations and Aubry set analysis.

Contribution

It provides a refined description of the limit behavior of eigenvalues and eigenfunctions, introducing a selection criterion based on Aubry set structure for the Hamilton-Jacobi limit.

Findings

Derived the limit of the principal eigenvalue.

Established a selection criterion for the limit eigenfunction.

Linked the spectral problem to a Hamilton-Jacobi framework.

Abstract

The paper deals with the Neumann spectral problem for a singularly perturbed second order elliptic operator with bounded lower order terms. The main goal is to provide a refined description of the limit behaviour of the principal eigenvalue and eigenfunction. Using the logarithmic transformation we reduce the studied problem to additive eigenvalue problem for a singularly perturbed Hamilton-Jacobi equation. Then assuming that the Aubry set of the Hamiltonian consists of a finite number of points or limit cycles situated in the domain or on its boundary, we find the limit of the eigenvalue and formulate the selection criterium that allows us to choose a solution of the limit Hamilton-Jacobi equation which gives the logarithmic asymptotics of the principal eigenfunction.