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

This paper deals with the Laplace equation in a bounded regular domain Ω of R N ( N ⩾ 2 ) coupled with a dynamical boundary condition of reactive–diffusive type. In particular we study the problem { Δ u = 0 in ( 0 , ∞ ) × Ω , u t = k u ν + l Δ Γ u on ( 0 , ∞ ) × Γ , u ( 0 , x ) = u 0 ( x ) on Γ , where u = u ( t , x ) , t ⩾ 0 , x ∈ Ω , Γ = ∂ Ω , Δ = Δ x denotes the Laplacian operator with respect to the space variable, while Δ Γ 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 u 0 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.