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

Abstract

Let $\Omega^{i}$, $\Omega^{o}$ be bounded open connected subsets of
${\mathbb{R}}^{n}$ that contain the origin. Let
$\Omega(\epsilon)\equiv
\Omega^{o}\setminus\epsilon\overline{\Omega^i}$ 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 $n\geq 3$.