On principal eigenvalues for periodic parabolic Steklov problems. (12th September 2002) - Godoy, T.; Dozo, E. Lami; Paczka, S.
Abstract : LetΩ be aC 2 + γ domain inℝ N, N ≥ 2, 0 < γ < 1 . LetT > 0 and letL be a uniformly parabolic operatorL 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 × ℝ, areT periodic int and satisfy some regularity assumptions. LetA be theN × N matrix whosei, j entry isa i j and letν be the unit exterior normal to∂ Ω . Letm be aT -periodic function (that may change sign) defined on∂ Ω whose restriction to∂ Ω × ℝ belongs toW q 2 − 1 / q, 1 − 1 / 2 q ( ∂ Ω × ( 0, T ) ) for some large enoughq . In this paper, we give necessary and sufficient conditions onm for the existence of principal eigenvalues for the periodic parabolic Steklov problemL u = 0 onΩ × ℝ, 〈 A ∇ u, ν 〉 = λ m u on∂ Ω × ℝ, u ( x, t ) = u ( x, t + T ), u > 0 onΩ × ℝ . Uniqueness and simplicity of the positive principal eigenvalue is proved and a related maximum principle is given.