Asymptotics for the eigenelements of the Neumann spectral problem with concentrated masses
J. Cainzos, Eugenia Pérez, M. Vilasanchez · Indiana University Mathematics Journal · 2007
We consider a spectral Neumann problem for the Laplace operator posed in a domain Ω of R 3 . We assume that the density function takes the value e -m in the small region EB C Q and the value 1 outside. e is a small parameter, e ∈ (0,1), and m is a strictly positive parameter; eB is the concentrated mass. We study the asymptotic behavior, as e → 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 Q, in which for m ≥ 3 the limit problem for the low frequencies is a 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 ∂Ω since the associated eigenfunctions lose in some way the local character affecting the concentrated mass.