Asymptotic approximation of eigenelements of the Dirichlet problem for the Laplacian in a domain with shoots

摘要

We study the asymptotic behavior of the eigenelements of the Dirichlet problem for the Laplacian in a two-dimensional bounded domain with thin shoots, depending on a small parameter epsilon. Under the assumption that the width of the shoots goes to zero, as epsilon tends to zero, we construct the limit (homogenized) problem and prove the convergence of the eigenvalues and eigenfunctions to the eigenvalues and eigenfunctions of the limit problem, respectively. Under the additional assumption that the shoots, in a fixed vicinity of the basis, are straight and periodic, and their width and the distance between the neighboring shoots are of order epsilon, we construct the two-term asymptotics of the eigenvalues of the problem, as epsilon -> 0.