Long time approximations for solutions of wave equations associated with the Steklov spectral homogenization problems

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 e, 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 e of size O(e) periodically distributed along the boundary; e also measures the periodicity of the structure. We consider associated second-order evolution problems on spaces of traces that depend on e, and we provide estimates for the time t in which standing waves, constructed from quasimodes, approach their solutions u e (t) as e → 0.

On multi-scale asymptotic structure of eigenfunctions in a boundary value problem with concentrated masses near the boundary

“…The verification of the equivalence is of great importance because there exist many examples (cf., e.g., [33]) when a substitution in a problem with corner singularities leads to incorrect solutions. It is also of great importance when dealing with evolution problems (cf., e.g., [19]). In addition, notice that infinitely many realizations of an elliptic problem as a self-adjoint operator with the discrete spectra occur in domains with corners (cf.…”

Boundary homogenization with large reaction terms on a strainer-type wall

“…We mention some of the first works in which keywords such as critical sizes and critical relations between parameters have been introduced [7,29,30] and [35], also [8] for nonhomogeneous boundary conditions. Let us refer to [5,6] and references therein for rapidly alternating Dirichlet-Steklov boundary conditions and [11,18,28] for further references and possible applications in the framework of Geophysics and Winkler beds (foundations). See [9][10][11][12][13][14][15] and [32] for an extensive and updated bibliography on different boundary homogenization problems with Robin-type boundary conditions.…”

Asymptotic analysis of the high frequencies for the Laplace operator in a thin T-like shaped structure

“…We explicitly notice that the low frequencies are related only with longitudinal vibrations along each branch of the T-like shaped thin structure (see Theorem 2.1 and Figure 2 (a)), while our results capture also vibrations depending on both variables, which are referred to as transverse vibrations along each branch, and are produced by high frequencies (see Theorems 5.2-5.4, Remark 5.5 and Figure 2 (b)). In fact, we get results for eigenvalues and eigenfunctions of (1.1) which are of interest in terms of the associated evolution problems since, from (1.8), we can construct standing waves which approach time-dependent solutions for long times, and these times can be precisely computed in terms of bounds for discrepancies such as that in (1.5) (see [41] and [34] for an abstract framework as well as for applications to very different vibrating systems). It should be emphasized that it seems to be a common fact to many mechanical systems arising in thin structures that the low frequencies give rise to longitudinal vibrations while for other kinds of vibrations such as torsional or stretching vibrations one must look among those associated to the high frequencies: see, for instance, [8], [23], [24], [35] and references therein.…”