AMBIENTUM BIOETHICA BIOLOGIA CHEMIA DIGITALIA DRAMATICA EDUCATIO ARTIS GYMNAST. ENGINEERING EPHEMERIDES EUROPAEA GEOGRAPHIA GEOLOGIA HISTORIA HISTORIA ARTIUM INFORMATICA IURISPRUDENTIA MATHEMATICA MUSICA NEGOTIA OECONOMICA PHILOLOGIA PHILOSOPHIA PHYSICA POLITICA PSYCHOLOGIA-PAEDAGOGIA SOCIOLOGIA THEOLOGIA CATHOLICA THEOLOGIA CATHOLICA LATIN THEOLOGIA GR.-CATH. VARAD THEOLOGIA ORTHODOXA THEOLOGIA REF. TRANSYLVAN
Rezumat articol ediţie STUDIA UNIVERSITATIS BABEŞ-BOLYAI
În partea de jos este prezentat rezumatul articolului selectat. Pentru revenire la cuprinsul ediţiei din care face parte acest articol, se accesează linkul din titlu. Pentru vizualizarea tuturor articolelor din arhivă la care este autor/coautor unul din autorii de mai jos, se accesează linkul din numele autorului.
Autori: TUĞBA AKYEL, MASSIMO LANZA DE CRISTOFORIS.
DOI: 10.24193/subbmath. 2022.2.14
Published Online: 2022-06-10
Published Print: 2022-06-30
pp. 383-402
VIEW PDF
FULL PDF
Abstract: Let $Omega^{i}$, $Omega^{o}$ be bounded open connected subsets of ${mathbb{R}}^{n}$ that contain the origin. Let $Omega(epsilon)equiv Omega^{o}setminusepsilonoverline{Omega^i}$ 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 $partialOmega^o$. Under appropriate conditions on the wave numbers in $epsilon Omega^i$ and $Omega(epsilon)$ and on the parameters involved in the transmission conditions on $epsilon partialOmega^i$, the transmission problem has a unique solution $(u^i(epsilon,cdot), u^o(epsilon,cdot))$ for small values of $epsilon>0$. Here $u^i(epsilon,cdot) $ and $u^o(epsilon,cdot) $ solve the Helmholtz equation in $epsilon Omega^i$ and $Omega(epsilon)$, respectively. Then we prove that if $xiinoverline{Omega^i}$ and $xiin mathbb{R}^nsetminus Omega^i $ then the rescaled solutions $u^i(epsilon,epsilonxi) $ and $u^o(epsilon,epsilonxi)$ can be expanded into a convergent power expansion of $epsilon$, $kappa_nepsilonlogepsilon$, $delta_{2,n}log^{-1}epsilon$, $ kappa_nepsilonlog^2epsilon $ for $epsilon$ small enough. Here $kappa_{n}=1$ if $n$ is even and $kappa_{n}=0$ if $n$ is odd and $delta_{2,2}equiv 1$ and $delta_{2,n}equiv 0$ if $ngeq 3$.
Key words: Helmholtz equation; microscopic behavior; real analytic continuation; singularly perturbed domain; transmission problem. Condition (FFC), Sobolev spaces, uncertainty quantification.
Mathematics Subject Classification (2010): 35J05, 35R30 41A60, 45F15, 47H30,78A30.
