On vibrating thin membranes with mass concentrated near the boundary: An asymptotic analysis

In a smooth bounded domain Omega of BbbR2 we consider the spectral problem - Delta uarepsilon = lambda (arepsilon ) ho arepsilon uarepsilon with boundary conditionpartial upartial uarepsilon = 0. The factor ho arepsilon plays the role of a mass density, and it is equal to a constant of order arepsilon-1 in an arepsilon -neighborhood of the boundary and to a constant of order arepsilon in the rest of Omega . We study the asymptotic behavior of the eigenvalues lambda (arepsilon ) and the eigenfunctions uarepsilon as arepsilon tends to zero. We obtain explicit formulas for the first and second terms of the corresponding asymptotic expansions by exploiting the solutions of certain auxiliary boundary value problems.