Long time approximations for solutions of wave equations associated with the Steklov spectral homogenization problems |
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Authors: | Miguel Lobo M Eugenia Pérez |
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Institution: | 1. Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avenida de los Castros s/n, 39005 Santander, Spain;2. Departamento de Matemática Aplicada y Ciencias de la Computación, E. T. S. I. Caminos, Canales y Puertos Universidad de Cantabria, Avenida de los Castros s/n, 39005 Santander, Spain |
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Abstract: | 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 ?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 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. |
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Keywords: | homogenization spectral analysis asymptotic analysis standing waves |
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