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On homogenization of the solutions of a boundary value problem for the Laplace equation in a partially perforated domain with Dirichlet conditions on the boundary of the cavities. (English. Russian original) Zbl 0848.35008
Russ. Acad. Sci., Dokl., Math. 50, No. 1, 47-52 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 337, No. 2, 168-171 (1994).
The paper deals with the asymptotic behaviour of solutions of a Laplace equation in a domain \(\Omega\subset \mathbb{R}^n\) perforated on one side \(\Omega^+= \Omega\cap \{x: x_1> 0\}\) with a Dirichlet condition on the boundary of the cavities. Specifically, let \(Q= (0, 1)^n\) be the unit cube and let \(Y_0\) be the “hole”, a domain with smooth boundary such that \(\overline Y_0\subset Q\). Let \[ G_\varepsilon= \mathbb{R}^n\backslash \bigcup_{m\in \mathbb{N}^n} \varepsilon(Y_0+ m) \] and let \(\Omega_\varepsilon= \Omega^+ \cap G_\varepsilon\), \(\Omega_\varepsilon= (\Omega\backslash \Omega^+)\cup \Omega_\varepsilon\). The boundary value problem is \[ \Delta u_\varepsilon+ f \quad\text{in} \quad \Omega_\varepsilon,\quad u_\varepsilon= 0 \quad \text{on}\quad \partial\Omega_\varepsilon \] with \(f\in C^\infty(\overline \Omega^+)\cap C^\infty(\overline{\Omega\backslash \Omega^+})\). An expansion \(u_\varepsilon= v_0(x)+ \varepsilon v_1(x)+\cdots\) is given, where the functions \(v_i\) solve auxiliary boundary value problems.
Reviewer: L.Ambrosio (Pavia)
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35C20 Asymptotic expansions of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25 Boundary value problems for second-order elliptic equations