"Microscopic behavior of the solutions of a transmission problem for the Helmholtz equation. A functional analytic approach"

"Let $\Omega^{i}$, $\Omega^{o}$ be bounded open connected subsets of ${\mathbb{R}}^{n}$ that contain the origin. Let $\Omega(\epsilon)\equiv \Omega^{o}\setminus\epsilon\overline{\Omega^i}$ 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 $\partial\Omega^o$. Under appropriate conditions on the wave numbers in $\epsilon \Omega^i$ and $\Omega(\epsilon)$ and on the parameters involved in the transmission conditions on $\epsilon \partial\Omega^i$, 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 $\xi\in\overline{\Omega^i}$ and $\xi\in \mathbb{R}^n\setminus \Omega^i$ then the rescaled solutions $u^i(\epsilon,\epsilon\xi) $ and $u^o(\epsilon,\epsilon\xi)$ can be expanded into a convergent power expansion of $\epsilon$, $\kappa_n\epsilon\log\epsilon$, $\delta_{2,n}\log^{-1}\epsilon$, $ \kappa_n\epsilon\log^2\epsilon $ for $\epsilon$ small enough. Here $\kappa_{n}=1$ if $n$ is even and $\kappa_{n}=0$ if $n$ is odd and $\delta_{2,2}\equiv 1$ and $\delta_{2,n}\equiv 0$ if $n\geq 3$."