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

Abstract

We consider a strongly elliptic differential expression of the form $b(D{)}^{\ast }g(x/\epsilon )b(D)$, $\epsilon >0$, 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 $b(D{)}^{\ast }g(x/\epsilon )b(D)$ in $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$ is denoted by $A_{\epsilon }$. 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 $A_{D,\epsilon }$ and $A_{N,\epsilon }$ given by $b(D{)}^{\ast }g(x/\epsilon )b(D)$ with the Dirichlet or Neumann boundary conditions, respectively. For the resolvents of the operators $A_{\epsilon }$, $A_{D,\epsilon }$, and $A_{N,\epsilon }$ in a regular point $\zeta$ we find approximations in different operator norms with error estimates depending on $\epsilon$ and the spectral parameter $\zeta$.

75 pages. arXiv admin note: text overlap with arXiv:1212.1148