Homogenization and correctors for the wave equation in non periodic perforated domains
The authors consider the homogenization of the wave equation in a (not necessarily periodically) perforated domain. They consider the wave equation \(\rho ^{\varepsilon }u_{\varepsilon }^{\prime \prime }-\text{div} (A^{\varepsilon }\nabla u_{\varepsilon })=f_{\varepsilon }\) in \(\Omega _{\varepsilon }\times (0,T)\), with the homogeneous Dirichlet boundary condition \(u_{\varepsilon }=0\) on \(\partial \Omega \times (0,T)\) and the homogeneous Neumann boundary condition \(A^{\varepsilon }\nabla u_{\varepsilon }\cdot \nu =0\) on \(\partial S_{\varepsilon }\times (0,T)\). Here \(\Omega \) is a bounded domain of \(\mathbb{R}^{n}\), \( S_{\varepsilon }\) is a collection of holes and \(\Omega _{\varepsilon }=\Omega \backslash S_{\varepsilon }\). The authors suppose that \( (S_{\varepsilon })_{\varepsilon }\) satisfies an ``admissible'' condition, in terms of the existence of linear (and uniformly continuous) extension operators belonging to \(\mathcal{L}(V_{\varepsilon },H_{0}^{1}(\Omega ))\) where \(V_{\varepsilon }=\{u\in H^{1}(\Omega _{\varepsilon }):u=0\) on \( \partial \Omega \}\). In the first part of their paper, the authors recall the definition of the \(H^{0}\)-convergence for sequences \((A^{\varepsilon },S_{\varepsilon })_{\varepsilon }\), previously defined by [\textit{M. Briane} et. al. Nonlinear partial differential equations and their applications. Collège de France seminar. Vol. XIII. Lectures held at the J. L. Lions seminar on applied mathematics, Paris, France, 1994--1996. Harlow: Longman. Pitman Res. Notes Math. Ser. 391, 62--100 (1998; Zbl 0943.35005)], and its compactness result. They then define the notion of global corrector and recall the existence of such a corrector in the present context, with its properties. As a first result, the authors prove the existence of an extension operator for every sequence \((S_{\varepsilon })_{\varepsilon }\) of admissible holes, in a stronger definition of this admissibility. They then build the convergence result for the (extended) solution of the wave equation. The main result of the paper states the existence of a corrector for this wave equation. The proof of the convergence result is obtained using the properties of the extension operator. The authors then prove the convergence of the energy associated to this wave equation. For the proof of the corrector result, the authors use the properties of a ``local'' test-function.
homogeneous Dirichlet boundary condition
scientific article; zbMATH DE number 5315160