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Authors: 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
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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.
