Abstract
This paper is concerned with an indefinite weight linear eigenvalue problem in cylindrical domains. We investigate the minimization of the positive principal eigenvalue under the constraint that the weight is bounded by a positive and a negative constant and the total weight is a fixed negative constant. Biologically, this minimization problem is motivated by the question of determining the optimal spatial arrangement of favorable and unfavorable regions for a species to survive. Both our analysis and numerical simulations for rectangular domains indicate that there exists a threshold value such that if the total weight is below this threshold value, then the optimal favorable region is a circular-type domain at one of the four corners, and a strip at the one end with shorter edge otherwise.
Highlights
Consider the following linear eigenvalue problem with indefinite weight (1.1)∆φ + λm(x)φ = 0 ∂φ=0 ∂n in Ω, on ∂Ω, where Ω is a bounded domain in RN with smooth boundary ∂Ω, n is the outward unit normal vector on ∂Ω, and the weight m is a bounded measurable function satisfying (1.2)−1 ≤ m(x) ≤ κ on Ω (κ > 0).We say that λ is a principal eigenvalue of (1.1) if λ has a positive eigenfunction φ ∈ H1(Ω)
We are interested in the dependence of the principal eigenvalue λ = λ(m) on the weight m
We focus on the case where Ω is a cylindrical domain given by
Summary
=0 ∂n in Ω, on ∂Ω, where Ω is a bounded domain in RN with smooth boundary ∂Ω, n is the outward unit normal vector on ∂Ω, and the weight m is a bounded measurable function satisfying (1.2). Since the species can be maintained if and only if ω > λ(m), we see that the smaller λ(m) is, the more likely the species can survive In this connection, the following question was raised and addressed by Cantrell and Cosner in [4, 5]: Among all functions m(x) that satisfy (A1), which m(x) will yield the smallest principal eigenvalue λ(m)? From the biological point of view, finding such a minimizing function m(x) is equivalent to determining the optimal spatial arrangement of the favorable and unfavorable parts of the habitat for species to survive [4, 5].
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