Homogenization of Elliptic Problems Depending on a Spectral Parameter

In $L_2({\mathbb R}^d;{\mathbb C}^n)$ we consider a strongly elliptic operator $A_\varepsilon$ given by the differential expression $b({\mathbf D})^*g({\mathbf x}/\varepsilon)b({\mathbf D})$, $\varepsilon >0$. Here $g({\mathbf x})$ is a bounded positive definite matrix-valued function assumed to be periodic with respect to some lattice and $b({\mathbf D})$ is a first-order differential operator. Let ${\mathcal O}\subset {\mathbb R}^d$ be a bounded domain with boundary of class $C^{1,1}$. We also study the operators $A_{D,\varepsilon}$ and $A_{N,\varepsilon}$ in $L_2({\mathcal O};{\mathbb C}^n)$ given by the same expression with Dirichlet or Neumann boundary conditions, respectively. We find approximations for the resolvents $(A_\varepsilon -\zeta I)^{-1}$, $(A_{D,\varepsilon} -\zeta I)^{-1}$, and $(A_{N,\varepsilon} -\zeta I)^{-1}$ in the operator ($L_2 \to L_2$)- and ($L_2 \to H^1$)-norms with error estimates depending on the parameters $\varepsilon$ and $\zeta$.