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

This paper is concerned with the periodic principal eigenvalue kλ(μ) associated with the operator − d 2 dx2 − 2λ d dx − μ(x)− λ, (1) where λ ∈ R and μ is continuous and periodic in x ∈ R. Our main result is that kλ(μ∗) ≤ kλ(μ), where μ∗ is the Schwarz rearrangement of the function μ. 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 μ∗ is the Steiner symmetrization of μ. For heterogeneous diffusion and advection, we prove that increasing the period of the coefficients decreases kλ and we compute the limit of kλ when the period of the coefficients goes to 0. Lastly, we prove that, in dimension 1, rearranging the diffusion term decreases kλ. These results rely on some new formula for the periodic principal eigenvalue of a nonsymmetric operator. Key-words: Schwarz rearrangement, eigenvalue optimization, nonsymmetric operator, reaction-diffusion equations AMS subject classification: 34L15, 35P15, 49R50, 47A75, 92D25, 92D40 Abbreviated title: Schwarz rearrangement of a nonsymmetric operator