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

• 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
• DOI: 10.1137/080743597

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.