Existence and Perturbation of Principal Eigenvalues for a Periodic-Parabolic Problem

Abstract

We give a necessary and sufficient condition for the existence of a positive principal eigenvalue for a periodic-parabolic problem with indefinite weight function. The condition was originally established by Beltramo and Hess [Comm. Part. Diff. Eq., 9 (1984), 919-941] in the framework of the Schauder theory of classical solutions. In the present paper, the problem is considered in the framework of variational evolution equations on arbitrary bounded domains, assuming that the coefficients of the operator and the weight function are only bounded and measurable. Finally, we establish a general perturbation theorem for the principal eigenvalues, which in particular allows quite singular perturbations of the domain. Motivation for the problem comes from population dynamics taking into account seasonal effects.