Asymptotic expansions of eigenvalues and eigenfunctions of a vibrating system with rigid light-weight inclusions (Q2870370)

The author studies spectral properties of a boundary value problem for second-order elliptic operator with singularly perturbed coefficients. The problem describes eigenmodes of an elastic system with a finite number of rigid light-weight inclusions of arbitrary shape. It is assumed that the ratio of rigidity coefficients of the inclusions and the main body has the order \(\varepsilon^{-1}\) as \(\varepsilon\to 0\), and the ratio of their mass densities is of order \(\varepsilon^{\varkappa}\) for \(\varkappa>0\). Complete asymptotic expansions of the eigenvalues and eigenfunctions of the problem are constructed and justified.