Nonlinear Differential Equations,

Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 51-67.

### Existence and perturbation of principal eigenvalues for a
periodic-parabolic problem

Daniel Daners

**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. We also establish a general perturbation
theorem for the principal eigenvalue, which in particular allows
quite singular perturbations of the domain. Motivation for the
problem comes from population dynamics taking into account seasonal
effects.
Published October 24, 2000.

Math Subject Classifications: 35K20, 35P05, 35B20, 47N20.

Key Words: principal eigenvalues, periodic-parabolic problems,
parabolic boundary-value problems, domain perturbation.

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Daniel Daners

Department of Mathematics, Brigham Young University

Provo, Utah 84602, USA

and:

School of Mathematics and Statistics, Univ. of Sydney

NSW~2006, Australia

e-mail: D.Daners@maths.usyd.edu.au

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