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

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...

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Bibliographic Details
Published in:SIAM journal on mathematical analysis Vol. 41; no. 6; pp. 2388 - 2406
Main Author: Nadin, Grégoire
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2010
Subjects:
Analysis of PDEs
Applied mathematics
Boundary conditions
Calculus of variations and optimal control
Cauchy problems
Eigenvalues
Exact sciences and technology
Hypotheses
Mathematical analysis
Mathematics
Nonlinear algebraic and transcendental equations
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Numerical methods in mathematical programming, optimization and calculus of variations
Numerical methods in optimization and calculus of variations
Optimization and Control
Population density
Sciences and techniques of general use
Symmetrization
eigenvalue optimization
nonsymmetric operator
Optimization method
Eigenvalue
34L15
47A75
Reaction diffusion equation
Optimization
92D25
Mathematical analysis
35P15
Population dynamics
Schwarz rearrangement
reaction-diffusion equations
92D40
49R50
Key-words: Schwarz rearrangement
reaction-diffusion equations AMS subject classification: 34L15
ISSN:0036-1410, 1095-7154
Online Access:Get full text
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Summary: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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:0036-1410
1095-7154
DOI:10.1137/080743597