ON VIBRATING THIN MEMBRANES WITH MASS CONCENTRATED NEAR THE BOUNDARY: AN ASYMPTOTIC ANALYSIS

In a smooth bounded domain Omega of R-2 we consider the spectral problem -Delta u(epsilon) = lambda(epsilon)rho(epsilon)u(epsilon) with boundary condition partial derivative u(epsilon)/partial derivative nu = 0. The factor rho(epsilon) plays the role of a mass density, and it is equal to a constant of order epsilon(-1) in an epsilon-neighborhood of the boundary and to a constant of order epsilon in the rest of Omega. We study the asymptotic behavior of the eigenvalues lambda(epsilon) and the eigenfunctions u(epsilon) as epsilon 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.