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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 $$t= 0$$.

##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35K05 Heat equation 35L05 Wave equation
##### Keywords:
convergence of the energy