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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11391/109283
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