On eigenvibrations of branched structures with heterogeneous mass density

Abstract

We deal with a spectral problem for the Laplace-Beltrami operator posed on a stratified set $\Omega$ which is composed of smooth surfaces joined along a line $\gamma$, the junction. Through this junction we impose the Kirchhoff-type vertex conditions, which imply the continuity of the solutions and some balance for normal derivatives, and Neumann conditions on the rest of the boundary of the surfaces. Assuming that the density is $O(\varepsilon^{-m})$ along small bands of width $O(\varepsilon)$, which collapse into the line $\gamma$ as $\varepsilon$ tends to zero, and it is $O(1)$ outside these bands, we address the asymptotic behavior, as $\varepsilon\to 0$, of the eigenvalues and of the corresponding eigenfunctions for a parameter $m\geq 1$. We also study the asymptotics for high frequencies when $m\in(1,2)$.