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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Vydáno v:	SIAM journal on mathematical analysis Ročník 41; číslo 6; s. 2388 - 2406
Hlavní autor:	Nadin, Grégoire
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2010
Témata:	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
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Shrnutí:	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.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 content type line 14
ISSN:	0036-1410 1095-7154
DOI:	10.1137/080743597