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

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Nadin, Grgoire
Copyright Year	2010
Abstract	This paper is concerned with the periodic principal eigenvalue $k_\lambda(\mu)$ associated with the operator $-\frac{d^2}{dx^2}-2\lambda\frac{d}{dx}-\mu(x)-\lambda^2$, where $\lambda\in\mathbb{R}$ and $\mu$ is continuous and periodic in $x\in\mathbb{R}$. Our main result is that $k_\lambda(\mu^)\leq k_\lambda(\mu)$, 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. Last, 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.
Starting Page	2388
Ending Page	2406
Page Count	19
File Format	PDF
ISSN	00361410
DOI	10.1137/080743597
e-ISSN	10957154
Journal	SIAM Journal on Mathematical Analysis (SJMAAH)
Issue Number	6
Volume Number	41
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2010-01-20
Access Restriction	Subscribed
Subject Keyword	Estimation of eigenvalues, upper and lower bounds eigenvalue optimization nonsymmetric operator Eigenvalue problems Population dynamics Ecology Schwarz rearrangement reaction-diffusion equations Eigenvalues, estimation of eigenvalues, upper and lower bounds
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Mathematics

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