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.
