Abstract and Applied Analysis
Volume 7 (2002), Issue 8, Pages 401-421
doi:10.1155/S1085337502204066
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 T. Godoy et al. 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.
Abstract
Let Ω be a C2+γ domain in ℝN,
N≥2, 0<γ<1. Let T>0 and let L be a uniformly
parabolic operator Lu=∂u/∂t−∑i,j (∂/∂xi) (aij(∂u/∂xj))+∑jbj (∂u/∂xi)+a0u, a0≥0, whose coefficients, depending on
(x,t)∈Ω×ℝ, 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×ℝ belongs to
Wq2−1/q,1−1/2q(∂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×ℝ,
〈A∇u,ν〉=λmu on
∂Ω×ℝ, u(x,t)=u(x,t+T), u>0 on
Ω×ℝ. Uniqueness and simplicity of the
positive principal eigenvalue is proved and a related maximum
principle is given.