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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).

##### MSC:
 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