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Correctors for the homogenization of the wave and heat equations. (English) Zbl 0837.35016
Summary: This paper is mainly devoted to the study of the corrector for the homogenization of the wave equation \[ \rho^\varepsilon u^\varepsilon_{tt}- \text{div}(A^\varepsilon\text{ grad } u^\varepsilon)= 0\qquad\text{in}\quad \Omega\times (0, T), \] \[ u^\varepsilon= 0\quad\text{on}\quad \partial\Omega\times (0, T),\quad u^\varepsilon(0)= a^\varepsilon,\quad u^\varepsilon_t(0)= b^\varepsilon\quad\text{in}\quad \Omega. \] A by now standard argument permits to pass to the limit in this equation and to obtain the homogenized equation satisfied by the limit \(u\) of \(u^\varepsilon\). Note however that the energy \(E^\varepsilon\) corresponding to \(u^\varepsilon\), defined by \[ E^\varepsilon= {1\over 2} \int_\Omega [\rho^\varepsilon |u^\varepsilon_t|^2+ A^\varepsilon\text{ grad } u^\varepsilon\text{ grad } u^\varepsilon] (x, t) dx= {1\over 2} \int_\Omega [\rho^\varepsilon |b^\varepsilon|^2+ A^\varepsilon\text{ grad } a^\varepsilon\text{ grad } a^\varepsilon] (x)dx \] does not in general converge to the energy corresponding to \(u\). We thus partition \(u^\varepsilon\) into a sum of two terms \(u^\varepsilon= \widetilde u^\varepsilon+ v^\varepsilon\). The first terms \(\widetilde u^\varepsilon\) solves the same wave equation with initial conditions \(\widetilde a^\varepsilon\) and \(\widetilde b^\varepsilon\) designed in a manner such that the energy \(\widetilde E^\varepsilon\) corresponding to \(\widetilde u^\varepsilon\) converges to \(E^0\). A corrector result for \(\widetilde u^\varepsilon\) can thus be proved, namely, \(\widetilde u^\varepsilon_t- u_t\to 0\) strongly in \(C^0([0, T]; L^2(\Omega))\), \(\text{grad } \widetilde u^\varepsilon- P^\varepsilon\text{ grad } u\to 0\) strongly in \(C_0([0, T]; (L^1(\Omega))^N)\). As far as \(v^\varepsilon\) is concerned, we prove that \(v^\varepsilon\) tends to zero weakly-\(*\) in \(L^\infty(0, T; H^1_0(\Omega))\cap W^{1,\infty} (0, T; L^2(\Omega))\). This convergence is strong if and only if \(a^\varepsilon- \widetilde a^\varepsilon\) and \(b^\varepsilon- \widetilde b^\varepsilon\) tend strongly to zero in \(H^1_0(\Omega)\) and in \(L^2(\Omega)\) respectively. If such is not the case \((1/2) \int_\varepsilon \rho^\varepsilon|v^\varepsilon|^2(x, t)dx\) and \((1/2) \int_\Omega (A^\varepsilon\text{ grad } v^\varepsilon\text{ grad } v^\varepsilon)(x, t)dx\) converge (in the weak-\(*\) topology of \(L^\infty(0, T)\)) to a positive constant. Thus \(v^\varepsilon\) is a perturbation which permeates all times.
The corrector problem for the heat equation is also investigated in this paper, in which case \(v^\varepsilon\) is proved to be an initial- boundary layer concentrated about the time \(t= 0\).

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K05 Heat equation
35L05 Wave equation