On the Laplace equation with dynamical boundary conditions of reactive-diffusive type

Abstract

This paper deals with the Laplace equation in a bounded regular domain Ω of RN (N ≥ 2) coupled with a dynamical boundary condition of reactive-diffusive type. In particular we study the problem{(Δ u = 0, in (0, ∞) × Ω,; ut = k uν + l ΔΓ u, on (0, ∞) × Γ,; u (0, x) = u0 (x), on Γ,) where u = u (t, x), t ≥ 0, x ∈ Ω, Γ = ∂ Ω, Δ = Δx denotes the Laplacian operator with respect to the space variable, while ΔΓ denotes the Laplace-Beltrami operator on Γ, ν is the outward normal to Ω, and k and l are given real constants. Well-posedness is proved for any given initial distribution u0 on Γ, together with the regularity of the solution. Moreover the Fourier method is applied to represent it in term of the eigenfunctions of a related eigenvalue problem. © 2009 Elsevier Inc. All rights reserved.