Microscopic behavior of the solutions of a transmission problem for the Helmholtz equation: a functional analytic approach
Küçük Resim Yok
Tarih
2022
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Stud. Univ. Babe¸s-Bolyai Math.
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Let Omega(i), Omega(o) be bounded open connected subsets of R-n that contain the origin. Let Omega(epsilon) equivalent to Omega(o) \ c (Omega) over bar (i) for small epsilon > 0. Then we consider a linear transmission problem for the Helmholtz equation in the pair of domains epsilon Omega(i) and Omega(epsilon) with Neumann boundary conditions on partial derivative Omega(o). Under appropriate conditions on the wave numbers in epsilon Omega(i) and Omega(epsilon) and on the parameters involved in the transmission conditions on epsilon partial derivative Omega(i), the transmission problem has a unique solution (u(i)(epsilon, .); u(o) (epsilon, .)) for small values of epsilon > 0. Here u(i)(epsilon, .) and u(o) (epsilon, .) solve the Helmholtz equation in epsilon Omega(i) and Omega(epsilon), respectively. Then we prove that if xi is an element of(Omega(i)) over bar and xi is an element of R-n\Omega(i) then the rescaled solutions u(i) (epsilon, epsilon xi) and u(o) (epsilon, epsilon xi) can be expanded into a convergent power expansion of epsilon, kappa(n) is an element of log epsilon, delta(2,n) log(-1) epsilon, kappa(n) is an element of log(2) epsilon for epsilon small enough. Here kappa(n) = 1 if n is even and kappa(n) = 0 if n is odd and delta(2,2) equivalent to 1 and delta(2,n) equivalent to 0 if n >= 3.
Açıklama
Anahtar Kelimeler
Helmholtz equation, Microscopic behavior, Real analytic continuation, Singularly perturbed domain, Transmission problem
Kaynak
Stud. Univ. Babeş-Bolyai Math.
WoS Q Değeri
N/A
Scopus Q Değeri
Cilt
67
Sayı
10
Künye
Akyel, T. and Lanza de Cristoforis, M (2022). Microscopic behavior of the solutions of a transmission problem for the Helmholtz equation: a functional analytic approach. Stud. Univ. Babeş-Bolyai Math., 67(10), p.383-402.
