On principal eigenvalues for periodic parabolic Steklov problems

Let Ω be a C2+γ domain in ℝN, N ≥ 2, 0 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 Wq21112−/q,−/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 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.