On the monotonicity property of the generalized eigenvalue for weakly-coupled cooperative elliptic systems

TL;DR

This paper investigates the monotonicity properties of the generalized principal eigenvalue in weakly-coupled cooperative elliptic systems, linking it to recurrence, stability, and stochastic representations of eigenfunctions.

Contribution

It establishes new equivalences between eigenvalue monotonicity, recurrence, minimal growth, and stability properties in elliptic systems.

Findings

Monotonicity on the right is equivalent to recurrence of the twisted operator.

Strict monotonicity corresponds to exponential stability of the twisted operators.

Monotonicity is also equivalent to the stochastic representation of the eigenfunction.

Abstract

We consider general linear non-degenerate weakly-coupled cooperative elliptic systems and study certain monotonicity properties of the generalized principal eigenvalue in $\mathbb{R}^d$ with respect to the potential. It is shown that monotonicity on the right is equivalent to the recurrence property of the twisted operator which is, in turn, equivalent to the minimal growth property at infinity of the principal eigenfunctions. The strict monotonicity property of the principal eigenvalue is shown to be equivalent with the exponential stability of the twisted operators. An equivalence between the monotonicity property on the right and the stochastic representation of the principal eigenfunction is also established.