Topological interpretation of averaged coefficients for periodically punctured domains. (Q1432169)

Given a bounded open set \(\Omega\) of \({\mathbb R}^3\) and \(F_1\) a \([0,1]^3\)-periodic set, define \(F_{\varepsilon} = \{\varepsilon x \;| \;x \in F_1\}\) and consider \(\Omega_{\varepsilon} := \Omega \cap F_{\varepsilon}\). The author studies the following system of equations: (\(u_{\varepsilon}\) a vector function \((u_1^{\varepsilon}, u_2^{\varepsilon},u_3^{\epsilon})\) and \(p_{\varepsilon}\) a scalar function) \( (u_{\varepsilon})_t + \nabla p_{\varepsilon} = f\) and \(\gamma (p_{\varepsilon})_t +\) div \(u_{\varepsilon} = 0\) on \(\Omega_{\varepsilon} \times (0,T)\), \((u_{\varepsilon}, \nu_{\varepsilon}) = 0\) on \(\partial\Omega_{\varepsilon} \times (0,T)\), \(u_{\varepsilon}(\cdot, 0) = 0\) and \(p_{\varepsilon}(\cdot, 0) = p_0\) on \(\Omega_{\varepsilon}\), with \(f \in L^2(0,T; L^2(\Omega_{\epsilon}))\), \(p_0 \in L^2(\Omega)/{\mathbb R}\), \(\nu_{\varepsilon}\) the outer normal to \(\partial \Omega_{\varepsilon}\), \(T, \gamma\) positive constants. This system models the propagation of small perturbations of a compressible fluid in a porous medium. The author states that there is a non-negative and symmetric matrix \(A\) such that \(u_{\varepsilon} \to u\) weakly-\(\ast\) in \(L^{\infty}(0,T; L^2(\Omega)^3)\) and \(p_{\varepsilon} \to \theta p\) weakly-\(\ast\) in \(L^{\infty}(0,T; L^2(\Omega)/{\mathbb R})\), \(\theta =\text{meas}\;( [0,1]^3 \cap F_1)\), where \(u_t - A(f -\nabla p) = 0\) and \(\gamma p_{tt} -\text{div} A (\nabla u - f) = 0\) on \(\Omega \times (0,T)\), \((A(f -\nabla p), \nu) = 0\) on \(\partial\Omega \times (0,T)\), \(p(\cdot, 0) = p_0\) and \(p_t(\cdot, 0) = 0\) on \(\Omega\), \(\nu\) the outer normal to \(\partial \Omega\). The main result is the following: the matrix \(A\) has rank \(3\) if and only if the set \(F_1\) is connected (the author says that the result is valid for every dimension \(n\) and not only in \({\mathbb R}^3\)). The interpretation is the following: if \(F_1\) is not connected there are directions in which waves do not propagate and rank \(A \leqslant 2\).