On principal eigenvalues for periodic parabolic Steklov problems

Abstract

Let Ω be a C2+γ domain in ℝN, N≥2, 00 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.