Microscopic behavior of the solutions of a transmission problem for the Helmholtz equation. A functional analytic approach
DOI:
https://doi.org/10.24193/subbmath.2022.2.14Keywords:
Helmholtz equation, microscopic behavior, real analytic continuation, singularly perturbed domain, transmission problem.Abstract
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.
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