Asymptotic behavior of the solutions of a transmission problem for the Helmholtz equation: A functional analytic approach

摘要

Let Ωi,  Ωo be bounded open connected subsets of ℝn that contain the origin. Let Ω(ϵ)≡ΩoϵΩi‾ for small ϵ > 0. Then, we consider a linear transmission problem for the Helmholtz equation in the pair of domains ϵΩi and Ω(ϵ) with Neumann boundary conditions on ∂Ωo. Under appropriate conditions on the wave numbers in ϵΩi and Ω(ϵ) and on the parameters involved in the transmission conditions on ϵ∂Ωi, the transmission problem has a unique solution (ui(ϵ, ·), uo(ϵ, ·)) for small values of ϵ > 0. Here, ui(ϵ, ·) and uo(ϵ, ·) solve the Helmholtz equation in ϵΩi and Ω(ϵ), respectively. Then, we prove that if x ∈ Ωo  {0}, then uo(ϵ, x) can be expanded into a convergent power expansion of ϵ, κnϵlogϵ,δ2,nlog−1ϵ for ϵ small enough. Here, κn=1 if n is even and κn=0 if n is odd, and δ2, 2 ≡ 1 and δ2, n ≡ 0 if n ≥ 3.