Homogenization of elliptic problems: error estimates in dependence of the spectral parameter

Abstract

We consider a strongly elliptic differential expression of the form $b(D)^* g(x/\varepsilon) b(D)$, $\varepsilon >0$, where $g(x)$ is a matrix-valued function in ${\mathbb R}^d$ assumed to be bounded, positive definite and periodic with respect to some lattice; $b(D)=\sum_{l=1}^d b_l D_l$ is the first order differential operator with constant coefficients. The symbol $b(\xi)$ is subject to some condition ensuring strong ellipticity. The operator given by $b(D)^* g(x/\varepsilon) b(D)$ in $L_2({\mathbb R}^d;{\mathbb C}^n)$ is denoted by $A_\varepsilon$. Let ${\mathcal O} \subset {\mathbb R}^d$ be a bounded domain of class $C^{1,1}$. In $L_2({\mathcal O};{\mathbb C}^n)$, we consider the operators $A_{D,\varepsilon}$ and $A_{N,\varepsilon}$ given by $b(D)^* g(x/\varepsilon) b(D)$ with the Dirichlet or Neumann boundary conditions, respectively. For the resolvents of the operators $A_\varepsilon$, $A_{D,\varepsilon}$, and $A_{N,\varepsilon}$ in a regular point $\zeta$ we find approximations in different operator norms with error estimates depending on $\varepsilon$ and the spectral parameter $\zeta$.