Minimization of the Principal Eigenvalue for an Elliptic Boundary Value Problem with Indefinite Weight, and Applications to Population Dynamics

Abstract

This paper is concerned with an indefinite weight linear eigenvalue problem which is related with biological invasions of species. 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. For an arbitrary domain, it is shown that every global minimizer must be of ``bang-bang'' type. When the domain is an interval, it is proved that there are exactly two global minimizers, for which the weight is positive at one end of the interval and is negative in the remainder. The biological implication is that a single favorable region at one end of the habitat provides the best opportunity for the species to survive, and also that the least fragmented habitat provides the best chance for the population to maintain its genetic variability.