Vibrating Systems with Rigid Light-Weight Inclusions: Asymptotics of the Spectrum and Eigenspaces

Abstract

We study the asymptotic behavior of the eigenvalues and eigenfunctions of a singularly perturbed boundary-value problem for a second-order elliptic operator. The problem describes the eigenmodes of an elastic system with finite number of stiff light-weight inclusions of arbitrary shape. The leading terms of the asymptotic representation of eigenelements are constructed with regard for their multiplicity. The justification of the asymptotic formulas is based on the uniform resolvent convergence of a certain family of unbounded self-adjoint operators.