Premium
Long time approximations for solutions of wave equations associated with the Steklov spectral homogenization problems
Author(s) -
Lobo Miguel,
Pérez M. Eugenia
Publication year - 2009
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1256
Subject(s) - homogenization (climate) , mathematics , eigenfunction , eigenvalues and eigenvectors , bounded function , mathematical analysis , dirichlet boundary condition , dirichlet distribution , boundary value problem , wavenumber , physics , biodiversity , ecology , quantum mechanics , optics , biology
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 ℝ 2 , 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 T ε 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 u ε ( t ) as ε→0. Copyright © 2009 John Wiley & Sons, Ltd.