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 $\zeta$.
Categories: math.AP