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Oscillations and energy densities in the wave equation. (English) Zbl 0803.35010
From the introduction: We consider the following wave equation with Dirichlet boundary conditions: \[ \rho^ \varepsilon {\partial^ 2 u^ \varepsilon \over \partial t^ 2} - \text{div} (A^ \varepsilon \text{grad} u^ \varepsilon) = f \quad \text{in} \quad \Omega \times (0,T), \]
\[ u^ \varepsilon = 0 \quad \text{on} \quad \partial \Omega \times (0,T),\quad u^ \varepsilon (0) = a^ \varepsilon \quad \text{in} \quad \Omega,\quad {\partial u^ \varepsilon \over \partial t} (0) = b^ \varepsilon \quad \text{in} \quad \Omega, \] where \(u^ \varepsilon\) is the unknown field while the other quantities are given. Under “minimal” assumptions on the various data, it is possible to partition the solution \(u^ \varepsilon\) into \(u^ \varepsilon = \widetilde u^ \varepsilon + v^ \varepsilon\), where \(\widetilde u^ \varepsilon\) can be explicitly described from the only knowledge of the weak limit field \(u\) of \(u^ \varepsilon\) while \(v^ \varepsilon\) converges weakly-\(*\) to 0 in the appropriate topology as \(\varepsilon\) tends to zero. The field \(v^ \varepsilon\) contains a wealth of information which is lost in the limit process. In particular the energy density associated to \(v^ \varepsilon\), i.e., \[ d^ \varepsilon = {1\over 2} \left( \rho^ \varepsilon \left( {\partial v^ \varepsilon \over \partial t} \right)^ 2 + A^ \varepsilon \text{grad} v^ \varepsilon \text{grad} v^ \varepsilon \right) \] satisfies \(H^ \varepsilon {\buildrel {\text{def}} \over =} \int_ \Omega d^ \varepsilon (x,t) dx\) is independent of \(t\), and \(H^ \varepsilon @> \varepsilon \to 0>>H\), where \(H\) is a positive (and in general strictly positive) constant.
Our goal in the present paper is to further describe the local behaviour of \(d^ \varepsilon\) as \(\varepsilon\) tends to zero; for example we will aim at computing the measure limit \(d^ 0\) of \(d^ \varepsilon\). From a more physical standpoint we strive to understand the space-time localization properties of the part of the elastic energy that remains trapped during the homogenization process.

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
Full Text: DOI
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