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Abstract and Applied Analysis
Volume 7 (2002), Issue 8, Pages 401-421

On principal eigenvalues for periodic parabolic Steklov problems

1Facultad de Matemática, Astronomía y Física and CIEM - Conicet Universidad Nacional de Córdoba, Ciudad Universitaria, Córdoba 5000, Argentina
2Departement de Mathematique, Université Libre de Bruxelles, Campus Plaine 214, Bruxelles 1050, Belgium
3IAM-Conicet and Universidad de Buenos Aires Saavedra 15, 3er Piso, Buenos Aires 1083, Argentina

Received 1 March 2002

Copyright © 2002 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let Ω be a C2+γ domain in N, N2, 0<γ<1. Let T>0 and let L be a uniformly parabolic operator Lu=u/ti,j(/xi)(aij(u/xj))+jbj(u/xi)+a0u, a00, whose coefficients, depending on (x,t)Ω×, are T periodic in t and satisfy some regularity assumptions. Let A be the N×N matrix whose i,j entry is aij and let ν be the unit exterior normal to Ω. Let m be a T-periodic function (that may change sign) defined on Ω whose restriction to Ω× belongs to Wq21/q,11/2q(Ω×(0,T)) for some large enough q. In this paper, we give necessary and sufficient conditions on m for the existence of principal eigenvalues for the periodic parabolic Steklov problem Lu=0 on Ω×, Au,ν=λmu on Ω×, u(x,t)=u(x,t+T), u>0 on Ω×. Uniqueness and simplicity of the positive principal eigenvalue is proved and a related maximum principle is given.