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 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.
On the Laplace equation with dynamical boundary conditions of reactive-diffusive type
VITILLARO, Enzo
2009
Abstract
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 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
