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We consider the transmission eigenvalue problem for an impenetrable obstaclewith Dirichlet boundary condition surrounded by a thin layer of non-absorbinginhomogeneous material. We derive a rigorous asymptotic expansion for the firsttransmission eigenvalue with respect to the thickness of the thin layer. Ourconvergence analysis is based on a Max-Min principle and an iterative approachwhich involves estimates on the corresponding eigenfunctions. We provideexplicit expressions for the terms in the asymptotic expansion up to orderthree.

Asymptotic analysis of the transmission eigenvalue problem for a Dirichlet obstacle coated by a thin layer of non-absorbing media