On principal eigenvalues for periodic parabolic Steklov problems

Let Ω be a C 2 + γ domain in ℠N , N ≥ 2 , 0 < γ < 1 . Let T > 0 and let L be a uniformly parabolic operator L u = ∂ u / ∂ t − ∑ i , j   ( ∂ / ∂ x i )   ( a i j ( ∂ u / ∂ x j ) ) + ∑ j b j   ( ∂ u / ∂ x i ) + a 0 u , a 0 ≥ 0 , 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 a i j 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 W q 2 − 1 / q , 1 − 1 / 2 q ( ∂ 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 L u = 0 on Ω × ℠, 〈 A ∇ u , ν 〉 = λ m u 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.