On the first eigenpair of singularly perturbed operators with oscillating coefficients

TL;DR

This paper investigates the asymptotic behavior of the first eigenpair of a singularly perturbed elliptic operator with oscillating coefficients, using viscosity solutions to derive an effective problem and exploring non-uniqueness and higher order asymptotics.

Contribution

It introduces a novel approach to analyze the limit behavior of eigenpairs in oscillatory environments using viscosity solutions, and addresses non-uniqueness in the effective problem.

Findings

Derived the effective problem using viscosity solutions.

Identified non-uniqueness issues in the limit problem.

Constructed higher order asymptotics for the ground state.

Abstract

The paper deals with a Dirichlet spectral problem for a singularly perturbed second order elliptic operator with rapidly oscillating locally periodic coefficients. We study the limit behaviour of the first eigenpair (ground state) of this problem. The main tool in deriving the limit (effective) problem is the viscosity solutions technique for Hamilton-Jacobi equations. The effective problem need not have a unique solution. We study the non-uniqueness issue in a particular case of zero potential and construct the higher order term of the ground state asymptotics.