Donato, Patrizia; Faella, Luisa; Monsurrò, Sara Correctors for the homogenization of a class of hyperbolic equations with imperfect interfaces. (English) Zbl 1197.35029 SIAM J. Math. Anal. 40, No. 5, 1952-1978 (2009). The authors consider the following problem \[ \begin{cases} u_\varepsilon'' - \text{div}(A^\varepsilon\nabla u_\varepsilon)= f_\varepsilon & \text{ in } (\Omega_{1_\varepsilon}\times \Omega_{2_\varepsilon})\times (0,T), \\ [A^\varepsilon\nabla u_\varepsilon]\cdot n_{1_\varepsilon}=0 & \text{ in } \Gamma_\varepsilon\times (0,T),\\ A^\varepsilon\nabla u_{1_\varepsilon}\cdot n_1{_\varepsilon}= -\varepsilon^\gamma h^\varepsilon[u_\varepsilon] & \text{ in } \Gamma_\varepsilon\times (0,T), \\ u_\varepsilon= 0 & \text{ in } \partial\Omega \times (0,T), \\ u_\varepsilon(0) = U_\varepsilon^0 & \text{ in } \Omega, \\ u_\varepsilon'(0)= U_\varepsilon^1 & \text{ in } \Omega, \end{cases} \]where \(A^\varepsilon(x):= A (x/\varepsilon)\), \(h^\varepsilon(x) := h (x/\varepsilon)\), \(u_varepsilon:= (u_{1_\varepsilon},u_{2_\varepsilon})\) is defined in \(\Omega_{1_\varepsilon}\times\Omega_{2_\varepsilon}\), \([\cdot]\) denotes the jump through \(\Gamma^\varepsilon\), and \(n_{i_\varepsilon}\) denotes the unitary outward noral to \(\Omega_{i_\varepsilon}\), \(i=1,2\). The homogenization problem is studied in a previous paper for \(\gamma \in (-1,1]\); for \(\gamma =1\) a memory effect appears (the limit is given by a problem described by a PDE coupled with an ODE).In the present paper the authors, with some additional assumptions on the data, give a corrector result. Reviewer: Fabio Paronetto (Padova) Cited in 14 Documents MSC: 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics 35L20 Initial-boundary value problems for second-order hyperbolic equations Keywords:homogenization; correctors; hyperbolic equations; interface problem PDFBibTeX XMLCite \textit{P. Donato} et al., SIAM J. Math. Anal. 40, No. 5, 1952--1978 (2009; Zbl 1197.35029) Full Text: DOI