Asymptotic 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 Double-struck capital Rn that contain the origin. Let omega(epsilon)equivalent to omega o\epsilon 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 differential 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 differential omega(i), the transmission problem has a unique solution (u(i)(epsilon, center dot), u(o)(epsilon, center dot)) for small values of epsilon > 0. Here, u(i)(epsilon, center dot) and u(o)(epsilon, center dot) solve the Helmholtz equation in epsilon omega(i) and omega(epsilon), respectively. Then, we prove that if x is an element of omega(o) \ {0}, then u(o)(epsilon, x) can be expanded into a convergent power expansion of epsilon, kappa n epsilon log epsilon,delta 2,nlog-1 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.