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

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Akyel T., Lanza de Cristoforis M.

Mathematical Methods in the Applied Sciences, cilt.45, sa.9, ss.5360-5387, 2022 (SCI-Expanded, Scopus)

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 45 Sayı: 9
  • Basım Tarihi: 2022
  • Doi Numarası: 10.1002/mma.8111
  • Dergi Adı: Mathematical Methods in the Applied Sciences
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Applied Science & Technology Source, Communication Abstracts, Compendex, INSPEC, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Sayfa Sayıları: ss.5360-5387
  • Anahtar Kelimeler: asymptotic behavior, Helmholtz equation, 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 (Formula presented.) 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 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, (Formula presented.) for ϵ small enough. Here, (Formula presented.) if n is even and (Formula presented.) if n is odd, and δ2, 2 ≡ 1 and δ2, n ≡ 0 if n ≥ 3.