Principal eigenvalue for some elliptic operators with large drift: Neumann boundary conditions

TL;DR

This paper analyzes the asymptotic behavior of the principal eigenvalue of certain elliptic operators with large drift in 2D, revealing its dependence on the system's omega-limit set components.

Contribution

It provides a detailed description of the eigenvalue's asymptotics based on the system's limit sets, including stable points and cycles, under non-degeneracy assumptions.

Findings

Eigenvalue behavior is determined by omega-limit set components.

Results include stable fixed points, limit cycles, and saddle connections.

Degenerate cases are also discussed.

Abstract

The paper is concerned with the principal eigenvalue of some linear elliptic operators with drift in two dimensional space. We provide a refined description of the asymptotic behavior for the principal eigenvalue as the drift rate approaches infinity. Under some non-degeneracy assumptions, our results illustrate that these asymptotic behaviors are completely determined by some connected components in the omega-limit set of the system of ordinary differential equations associated with the drift term, which includes stable fixed points, stable limit cycles, hyperbolic saddles connecting homoclinic orbits, and families of closed orbits. Some discussions on degenerate cases are also included.