The Effect of the Schwarz Rearrangement on the Periodic Principal Eigenvalue of a Nonsymmetric Operator

TL;DR

This paper investigates how the Schwarz rearrangement affects the principal eigenvalue of a nonsymmetric operator, revealing implications for population invasion speed and extending results to higher dimensions and heterogeneous media.

Contribution

It establishes that Schwarz rearrangement decreases the principal eigenvalue in one dimension and explores related effects in higher dimensions and heterogeneous environments.

Findings

Schwarz rearrangement reduces the eigenvalue in 1D.

Increasing period length decreases the eigenvalue.

Rearranging diffusion decreases the eigenvalue in 1D.

Abstract

This paper is concerned with the periodic principal eigenvalue $k_\lambda(\mu)$ associated with the operator $- {d^2\over dx^2} - 2\lambda {d\over dx} - \mu(x) - \lambda^2$ , (1) where $\lambda\in \mathbb{R}$ and $\mu$ is continuous and periodic in $x\in\mathbb{R}$. Our main result is that $k_\lambda(\mu^*) \le k_\lambda(\mu)$, where $\mu^*$ is the Schwarz rearrangement of the function $\mu$. From a population dynamics point of view, using reaction-diffusion modeling, this result means that the fragmentation of the habitat of an invading population slows down the invasion. We prove that this property does not hold in higher dimension, if $\mu^*$ is the Steiner symmetrization of $\mu$. For heterogeneous diffusion and advection, we prove that increasing the period of the coefficients decreases $k_\lambda$ and we compute the limit of $k_\lambda$ when the period of the coefficients goes to 0. Lastly, we prove that, in dimension 1, rearranging the diffusion term decreases $k_\lambda$. These results rely on some new formula for the periodic principal eigenvalue of a nonsymmetric operator.