  We consider the transmission eigenvalue problem for an impenetrable obstacle
with Dirichlet boundary condition surrounded by a thin layer of non-absorbing
inhomogeneous material. We derive a rigorous asymptotic expansion for the first
transmission eigenvalue with respect to the thickness of the thin layer. Our
convergence analysis is based on a Max-Min principle and an iterative approach
which involves estimates on the corresponding eigenfunctions. We provide
explicit expressions for the terms in the asymptotic expansion up to order
three.
