Homogenization and corrector for the wave equation with discontinuous coefficients in time
The authors consider the wave equation \[ \partial_t(\rho_n(t,x) \partial_t u_n) -\text{div}_x(A_n(t,x)\nabla_x u_n)= F_n \quad\text{in }(0,T)\times \Omega. \] The authors assume that \(\rho_n\) and \(A_n\) are bounded in \(BV(0,T;L^\infty(\Omega ))\) and \(BV(0,T;L^\infty(\Omega ;M_N^s))\) respectively. Here \(M([0,T];X)\) denotes the space of bounded Borel measures from \([0,T]\) into \(X\) and \(M_N^s\) the space of square symmetric matrices of order \(N\). Homogeneous Dirichlet boundary conditions \(u_n=0\) are imposed on \((0,T)\times \partial \Omega \) and initial conditions are added both for \(u_n\) and \(\partial_tu_n: u_n(0,x)=u_n^0(x)\) and \(\rho_n(0,x)\partial_tu_n(0,x)=\vartheta_n^1(x)\) in \(\Omega \). The initial conditions \( u_n^0\) and \(\vartheta _n^1\) are supposed to be bounded in \( H_0^1(\Omega )\) and \(L^2(\Omega )\) respectively. The second member \( F_n\) of the wave equation is taken as \(F_n=f_n+g_n\) with \(f_n\) bounded in the space of measures \(M([0,T];L^2(\Omega ))\) and \(g_n\) bounded in \(BV(0,T;H^{-1}(\Omega ))\). Under these hypotheses, this problem has a unique solution and the authors have to impose that \(u_n\) and \( \partial _tu_n\) are bounded in \(L^\infty(0,T;H_0^1(\Omega ))\) and \(L^\infty(0,T;L^2(\Omega ))\) respectively. The main result of the paper establishes the asymptotic behaviour of the solution of this problem. The sequence \((u_n)_n\) converges weakly-* in \(L^\infty (0,T;H_0^1(\Omega ))\) to the solution of a similar problem where \( \rho _n\) is replaced by its weak-* limit \(\rho \) in \(L^\infty((0,T)\times \Omega )\) and \(A_n\) by the matrix \(A\) such that \(A(t,.)\) is the \(H\)-limit of \(A_n(t,.)\) for every \(t\in (0,T)\) up to a countable set. The authors first introduce a variational formulation for the original wave problem. They introduce the solutions of some cell elliptic problems and use the compactness result associated to the \(H\)-convergence in order to pass to the limit. The authors then build a corrector for this problem under some further hypotheses on the coefficients \(\rho_n\) and \(A_n\). The paper ends with a counterexample proving that the corrector which is built in the previous section is not valid when the coefficients \(A_n\) and \(\rho_n\) are only supposed to be bounded in \(C^1([0,T];L^\infty(\Omega))\) and \(C^1([0,T];L^\infty(\Omega;M_N))\) respectively.
scientific article; zbMATH DE number 5880880