Microscopic behavior of the solutions of a transmission problem for the Helmholtz equation. A functional analytic approach - AKYEL, Tuğba; LANZA DE CRISTOFORIS, Massimo (Studia Universitatis Babeș-Bolyai Mathematica, 2022)
Let Ωi, Ωo be bounded open connected subsets of Rn that contain the origin. Let Ω(E) ≡ Ωo EΩi for small E > 0. Then we consider a linear transmission problem for the Helmholtz equation in the pair of domains EΩi and Ω(E) with Neumann boundary conditions on ∂Ωo. Under appropriate conditions on the wave numbers in EΩi and Ω(E) and on the parameters involved in the transmission conditions on E∂Ωi, the transmission problem has a unique solution (ui(E, ·), uo(E, ·)) for small values of E > 0. Here ui(E, ·) and uo(E, ·) solve the Helmholtz equation in EΩi and Ω(E), respectively. Then we prove that if ξ ∈ Ωi and ξ ∈ Rn \Ωi then the rescaled solutions ui(E, Eξ) and uo(E, Eξ) can be expanded into a convergent power expansion of E, κnE log E, δ2,n log−1 E, κnE log2 E for E 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. Mathematics Subject Classification (2010): 35J05, 35R30 41A60, 45F15, 47H30, 78A30. Received 27 January 2022; Accepted 9 March 2022.