Microscopic behavior of the solutions of a transmission problem for the Helmholtz equation. A functional analytic approach
Studia Universitatis Babes-Bolyai Mathematica, cilt.67, sa.2, ss.383-402, 2022 (Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 67 Sayı: 2
- Basım Tarihi: 2022
- Doi Numarası: 10.24193/subbmath.2022.2.14
- Dergi Adı: Studia Universitatis Babes-Bolyai Mathematica
- Derginin Tarandığı İndeksler: Scopus
- Sayfa Sayıları: ss.383-402
- Anahtar Kelimeler: Helmholtz equation, Microscopic behavior, Real analytic continuation, Singularly perturbed domain, Transmission problem
- Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu
- Maltepe Üniversitesi Adresli: Evet
Özet
Let Ωi, Ωo be bounded open connected subsets of Rn that contain the origin. Let (Formula Presented) for small ε > 0. Then we consider a linear transmission problem for the Helmholtz equation in the pair of domains ε Ωi and Ω(ε) with Neumann boundary conditions on ∂Ωo. Under appropriate conditions on the wave numbers in ε Ωi and Ω(ε) and on the parameters involved in the transmission conditions on ε Ωi, the transmission problem has a unique solution (Formula Presented) for small values of ε > 0. Here ui(ε,.) and uo(ε,.) solve the Helmholtz equation in ε Ωi and Ω(ε), respectively. Then we prove that if (Formula Presented)and (Formula Presented) then the rescaled solutions (Formula Presented) and (Formula Presented) can be expanded into a convergent power expansion of (Formula Presented) for ε small enough. Here kn = 1 if n is even and kn = 0 if n is odd and δ2,2 ≡ 1 and δ2,n ≡ 0 if n ≥ 3.