IUMJ

Title: Asymptotics for the eigenelements of the Neumann spectral problem with concentrated masses

Authors: J. Cainzos, E. Perez and M. Vilasanchez

Issue: Volume 56 (2007), Issue 4, 1939-1987

Abstract:

We consider a spectral Neumann problem for the Laplace operator posed in a domain $\Omega$ of $\mathbb{R}^3$. We assume that the density function takes the value $\varepsilon^{-m}$ in the small region $\varepsilon B \subset \Omega$ and the value $1$ outside. $\varepsilon$ is a small parameter, $\varepsilon \in (0,1)$, and $m$ is a strictly positive parameter; $\varepsilon B$ is \emph{the concentrated mass}. We study the asymptotic behavior, as $\varepsilon \to 0$, of the eigenvalues and the corresponding eigenfunctions for $m > 2$. Low and high frequencies are considered, and additional information on the structure of the associated eigenfunctions is provided. We also consider the case of several concentrated masses inside the domain $\Omega$, in which for $m \geq 3$ the limit problem for the low frequencies is a \emph{non-local system of equations} in the microscopic variables, involving simultaneously all the domains in which the concentrated masses are placed. This strongly differs from the case where a Dirichlet condition is imposed on $\partial\Omega$ since the associated eigenfunctions lose in some way the local character affecting the concentrated mass.