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Homogenization and correctors for the heat equation in perforated domains. (English) Zbl 1102.35305
Let \(\Omega\) be a bounded domain in \(\mathbb R^n\), and let \(S\) be a fixed bounded open set. Let \(\Omega_\epsilon\) be the domain obtained by removing from \(\Omega\) a closed set \(S_\epsilon\), which is a union of disjoint translates of \(\epsilon\overline S\). Let \(A(y)\) be a given periodic, positive definite matrix satisfying certain conditions, and let \(A^\epsilon(x)=A(\frac{x}{\epsilon})\). The authors study the asymptotic behaviour as \(\epsilon \to 0\) of the solution \(u_\epsilon\) to the problem \[ u'_\epsilon -div(A^\epsilon \nabla u_\epsilon)=f_\epsilon \quad\quad\hbox{in} \quad\Omega \times]0,T[, \] with the boundary conditions \(u_\epsilon=0\) on \(\partial \Omega \times]0,T[, \quad(A^\epsilon \nabla u_\epsilon).\nu =0\) on \(\partial S_\epsilon \times]0,T[,\) and the initial condition \(u_\epsilon(.,0)=u^0_\epsilon\) in \(\Omega_\epsilon.\) Here \(f_\epsilon \in L^2(\Omega_\epsilon \times]0,T[), \quad u^0_\epsilon \in L^2(\Omega_\epsilon),\) and \(\nu\) is the outward unit normal to \(\partial\Omega_\epsilon.\) They show that \(u_\epsilon\) converges to the solution of a related problem, under appropriate weak convergence hypotheses on \(f_\epsilon\) and \(u^0_\epsilon.\)
The results have been announced in C. R. Acad. Sci., Paris, Sér. I 324, No. 7, 789–794 (1997; Zbl 0877.35014).

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K20 Initial-boundary value problems for second-order parabolic equations
35K05 Heat equation