Long time approximations for solutions of wave equations associated with the Steklov spectral homogenization problems
Miguel Lobo, María‐Eugenia Pérez‐Martínez · Mathematical Methods in the Applied Sciences · 2009
The interest in the use of quasimodes, or almost frequencies and almost eigenfunctions, to describe asymptotics for low-frequency and high-frequency vibrations in certain singularly perturbed spectral problems, which depend on a small parameter ε, has been recently highlighted in many papers. In this paper we deal with the low frequencies for a Steklov-type eigenvalue homogenization problem: we consider harmonic functions in a bounded domain of ℝ2, and strongly alternating boundary conditions of the Dirichlet and Steklov type on a part of the boundary. The spectral parameter appears in the boundary condition on small segments Tε of size O(ε) periodically distributed along the boundary; ε also measures the periodicity of the structure. We consider associated second-order evolution problems on spaces of traces that depend on ε, and we provide estimates for the time t in which standing waves, constructed from quasimodes, approach their solutions uε(t) as ε→0. Copyright © 2009 John Wiley & Sons, Ltd.