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Homogenization in some weakly connected domains. (English) Zbl 0928.35037

Summary: We study the homogenization of the second order elliptic equation \(-\text{div}(A_\varepsilon \nabla u_\varepsilon)+ b_\varepsilon u_\varepsilon=f\) with Neumann boundary condition, in a weakly connected domain \(\Omega_\varepsilon\). The set \(\Omega_\varepsilon\) is an open subset of a fixed bounded open set \(\Omega\) of \(\mathbb{R}^N\), \(N\geq 2\), which is made of two separated parts \(\Omega_{1\varepsilon}\) and \(\Omega_{2\varepsilon}\) jointed by a set of thin bridges \(\omega_\varepsilon\), namely \(\Omega_\varepsilon =\Omega_{1 \varepsilon}\cup \Omega_{2\varepsilon}\cup \omega_\varepsilon\); both domains \(\Omega_{1\varepsilon}\) and \(\Omega_{2\varepsilon}\) are \(\varepsilon\)-periodic while \(|\omega_\varepsilon|\to 0\). Three geometrical cases are studied according to the connectedness of \(\Omega_{1\varepsilon}\) and \(\Omega_{2 \varepsilon}\). We moreover distinguish three cases for the size of the bridges, whose the critical one corresponds to \(|\omega_\varepsilon |\sim c \varepsilon^2\). In each case, the homogenized problem that we obtain is a coupled system which is more or less coupled according to size of the bridges. A corrector result is also given, which yields a strong approximation of \(\nabla u_\varepsilon\). This corrector includes an interaction term in the critical size.

MSC:

35J25 Boundary value problems for second-order elliptic equations
74E05 Inhomogeneity in solid mechanics
74A40 Random materials and composite materials
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