A singularly perturbed spectral problem for a biharmonic operator with Neumann conditions

Abstract

We study a mathematical model of a composite plate that consists of two components with similar elastic properties but different distributions of density. The area of the domain occupied by one of the components is infinitely small as $ε → 0$. We investigate the asymptotic behavior of the eigenvalues and eigenfunctions of the boundary-value problem for a biharmonic operator with Neumann conditions as $ε → 0$. We describe four different cases of the limiting behavior of the spectrum, depending on the ratio of densities of the medium components. In particular, we describe the so-called Sanches-Palensia effect of local vibrations: A vibrating system has a countable series of proper frequencies infinitely small as $ε → 0$ and associated with natural forms of vibrations localized in the domain of perturbation of density.