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

TL;DR

This paper derives an asymptotic expansion for the first transmission eigenvalue of a Dirichlet obstacle coated with a thin non-absorbing layer, providing explicit terms up to third order.

Contribution

It introduces a rigorous asymptotic analysis of the transmission eigenvalue problem for obstacles with thin coatings, including explicit expansion terms.

Findings

Explicit asymptotic expansion up to third order for the first transmission eigenvalue.

Convergence analysis based on Max-Min principle and eigenfunction estimates.

Provides a framework for understanding eigenvalue behavior in coated obstacle problems.

Abstract

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.