Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients

Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. Let $0<\varepsilon\leqslant 1$. In $L_2(\mathcal{O};\mathbb{C}^n)$ we consider a positive definite strongly elliptic second-order operator $B_{D,\varepsilon}$ with Dirichlet boundary condition. Its coefficients are periodic and depend on $\mathbf{x}\varepsilon$. The principal part of the operator is given in factorized form, and the operator has lower order terms. We study the behavior of the generalized resolvent $(B_{D,\varepsilon}-\zeta Q_0(\cdot/\varepsilon))^{-1}$ as $\varepsilon \to 0$. Here the matrix-valued function $Q_0$ is periodic, bounded, and positive definite; $\zeta$ is a complex-valued parameter. We find approximations of the generalized resolvent in the $L_2(\mathcal{O};\mathbb{C}^n)$-operator norm and in the norm of operators acting from $L_2(\mathcal{O};\mathbb{C}^n)$ to the Sobolev space $H^1(\mathcal{O};\mathbb{C}^n)$ with two-parameter error estimates (depending on $\varepsilon$ and $\zeta$). Approximations of the generalized resolvent are applied to the homogenization of the solution of the first initial-boundary value problem for the parabolic equation $Q_0({\mathbf x}/\varepsilon)\partial_t {\mathbf v}_\varepsilon({\mathbf x},t)=- ( B_{D,\varepsilon} {\mathbf v}_\varepsilon)({\mathbf x},t)$.