Massimo Lanza de Cristoforis

Let Ω i , Ω o be bounded open connected subsets of that contain the origin. Let for small ϵ > 0. Then, we consider a linear transmission problem for the Helmholtz equation in the pair of domains ϵ Ω i and Ω( ϵ ) with Neumann boundary conditions on ∂ Ω o . Under appropriate conditions on the wave numbers in ϵ Ω i and Ω( ϵ ) and on the parameters involved in the transmission conditions on ϵ ∂ Ω i , the transmission problem has a unique solution ( u i ( ϵ , ·), u o ( ϵ , ·)) for small values of ϵ > 0. Here, u i ( ϵ , ·) and u o ( ϵ , ·) solve the Helmholtz equation in ϵ Ω i and Ω( ϵ ), respectively. Then, we prove that if x ∈ Ω o \ {0}, then u o ( ϵ , x ) can be expanded into a convergent power expansion of ϵ , for ϵ small enough. Here, if n is even and if n is odd, and δ 2, 2 ≡ 1 and δ 2, n ≡ 0 if n ≥ 3.