Annali di Matematica Pura ed Applicata

, Volume 188, Issue 2, pp 269–295 | Cite as

The principal eigenvalue of a space–time periodic parabolic operator

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Abstract

This paper deals with the generalized principal eigenvalue of the parabolic operator \({\mathcal{L}\phi = \partial_{t}\phi - \nabla \cdot(A(t, x)\nabla\phi) + q(t, x) \cdot \nabla\phi - \mu(t, x)\phi}\) , where the coefficients are periodic in t and x. We give the definition of this eigenvalue and we prove that it can be approximated by a sequence of principal eigenvalues associated to the same operator in a bounded domain, with periodicity in time and Dirichlet boundary conditions in space. Next, we define a family of periodic principal eigenvalues associated with the operator and use it to give a characterization of the generalized principal eigenvalue. Finally, we study the dependence of all these eigenvalues with respect to the coefficients.

Keywords

Generalized principal eigenvalue Parabolic periodic operator 

Mathematics Subject Classification (2000)

35P05 
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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Département de Mathématiques et ApplicationsÉcole Normale Supérieure, CNRS UMR8553Paris cedex 05France

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