Asymptotic Approximation of Eigenelements of the Dirichlet Problem for the Laplacian in a Junction with Highly Oscillating Boundary

Abstract

We study the asymptotic behavior of the eigenelements of the Dirichlet problem for the Laplacian in a bounded domain, a part of whose boundary, depending on a small parameter $\varepsilon$, is highly oscillating; the frequency of oscillations of the boundary is of order $\varepsilon$ and the amplitude is fixed. We construct and analyze second-order asymptotic approximations, as $\varepsilon \to 0$, of the eigenelements in the case of simple eigenvalues of the limit problem