On Homogenization for Piecewise Locally Periodic Operators

We discuss homogenization of a strongly elliptic operator \(\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon_\#)\nabla\) on a bounded \(C^{1,1}\) domain in \(\mathbb R^d\) with either Dirichlet or Neumann boundary condition. The function \(A\) is piecewise Lipschitz in the first variable and periodic in the second one, and the function \(\varepsilon_\#\) is identically equal to \(\varepsilon_i(\varepsilon)\) on each piece \(\Omega_i\), with \(\varepsilon_i(\varepsilon)\to0\) as \(\varepsilon\to0\). For \(\mu\) in a resolvent set, we show that the resolvent \((\mathcal A^\varepsilon-\mu)^{-1}\) converges, as \(\varepsilon\to0\), in the operator norm on \(L_2(\Omega)^n\) to the resolvent \((\mathcal A^0-\mu)^{-1}\) of the effective operator at the rate \(\varepsilon_ {\vee} \), where \(\varepsilon_ {\vee} \) stands for the largest of \(\varepsilon_i(\varepsilon)\). We also obtain an approximation for the resolvent in the operator norm from \(L_2(\Omega)^n\) to \(H^1(\Omega)^n\) with error of order \(\varepsilon_ {\vee} ^{1/2}\).