The periodic unfolding method for the wave equation in domains with Holes (Q2861076)

The main purpose of the paper is to define a periodic unfolding operator for evolution problems posed in perforated domains. Starting from a domain \( \Omega \) in \(\mathbb{R}^{n}\) with Lipschitz boundary \(\partial \Omega \), the authors first define the periodically perforated domain \(\Omega _{\varepsilon }^{\ast }=\Omega \setminus \cup _{\xi \in G}\varepsilon (\xi +S)\) where \(\xi \) means the points of the network with entire coordinates and \(S\subset \overline{Y}\) denotes the reference perforation with smooth boundary. The unfolding operator \(\mathcal{T}_{\varepsilon }^{\ast }\) for time-dependent functions is then defined for every function \(\phi \in L^{q}(0,T;L^{p}(\Omega _{\varepsilon }^{\ast }))\) through \(\mathcal{T} _{\varepsilon }^{\ast }(\phi )(x,y,t)=\phi (\varepsilon [ x/\varepsilon ]_{Y}+\varepsilon y,t)\) for a.e. \((x,y,t)\in \widehat{\Omega } _{\varepsilon }\times Y^{\ast }\times (0,T)\) and \(\mathcal{T}_{\varepsilon }^{\ast }(\phi )(x,y,t)=0\) for a.e. \((x,y,t)\in \Lambda _{\varepsilon }\times Y^{\ast }\times (0,T)\). Here the authors use the unique decomposition \(z=[z]_{Y}+\{z\}\) where \([z]_{Y}\) has entire coordinates and \( \{z\}\in Y\), \(\widehat{\Omega }_{\varepsilon }=\text{Interior}(\cup _{\xi }(\varepsilon (\xi +\overline{Y}))\), \(\Lambda _{\varepsilon }=\Omega \setminus \widehat{\Omega }_{\varepsilon }\) and \(Y^{\ast }=Y\setminus \overline{S}\). The authors here extend to a nonsteady situation the definition of the unfolding operator already introduced by \textit{D. Cioranescu} et al. [SIAM J. Math. Anal. 40, No. 4, 1585--1620 (2008; Zbl 1167.49013)]. Then the authors present and prove the properties of this unfolding operator. In the rest of their paper, the authors prove homogenization and corrector results for the wave equations in periodically perforated domains with a homogeneous Neumann condition on the holes, using this unfolding operator. They indeed consider the problem \(u_{\varepsilon }^{\prime \prime }-\text{div}(A^{\varepsilon }\nabla u_{\varepsilon })=f_{\varepsilon }\) in \(\Omega _{\varepsilon }^{\ast }\times (0,T)\) with the homogeneous Dirichlet boundary condition \(u_{\varepsilon }=0\) on \( \partial \Omega \times (0,T)\) and the homogeneous Neumann boundary condition \(A^{\varepsilon }\nabla u_{\varepsilon }\cdot n_{\varepsilon }=0\) on \( \partial S_{\varepsilon }\times (0,T)\). Initial data are imposed on \( u_{\varepsilon }\) and on \(u_{\varepsilon }^{\prime }\). Assuming that \( \mathcal{T}_{\varepsilon }^{\ast }A^{\varepsilon }\) converges strongly in \( (L^{1}(\Omega \times Y^{\ast }))^{n}\) to some fixed matrix \(A\) and appropriate hypotheses on the data, the authors describe the asymptotic behaviour of \(u_{\varepsilon }\) using the unfolding operator. They also prove the convergence of the energy and corrector results for this wave equation. Thus doing, they recover the homogenization results obtained by \textit{D. Cioranescu} and \textit{P. Donato} [J. Math. Pures Appl. (9) 68, No. 2, 185--213 (1989; Zbl 0627.35057)]. They also recover and complete the corrector results proved [\textit{A. Nabil}, GAKUTO Int. Ser., Math. Sci. Appl. 9, 309--321 (1995; Zbl 0912.35018)].