Homogenization of elliptic operators with periodic coefficients in dependence of the spectral parameter

Abstract:

Differential expressions of the form $b(\mathbf {D})^* g(\mathbf {x}/\varepsilon ) b(\mathbf {D})$, $\varepsilon >0$, are considered, where a matrix-valued function $g(\mathbf {x})$ in $\mathbb {R}^d$ is assumed to be bounded, positive definite, and periodic with respect to some lattice; $b(\mathbf {D})=\sum _{l=1}^d b_l D_l$ is a first order differential operator with constant coefficients. The symbol $b({\boldsymbol \xi })$ is subject to some condition ensuring strong ellipticity. The operator in $L_2(\mathbb {R}^d;\mathbb {C}^n)$ given by the expression $b(\mathbf {D})^* g(\mathbf {x}/\varepsilon ) b(\mathbf {D})$ is denoted by $\mathcal {A}_\varepsilon$. Let $\mathcal {O} \subset \mathbb {R}^d$ be a bounded domain of class $C^{1,1}$. The operators $\mathcal {A}_{D,\varepsilon }$ and $\mathcal {A}_{N,\varepsilon }$ under study are generated in the space $L_2(\mathcal {O};\mathbb {C}^n)$ by the above expression with the Dirichlet or Neumann boundary conditions. Approximations in various operator norms for the resolvents $(\mathcal {A}_\varepsilon - \zeta I)^{-1}$, $(\mathcal {A}_{D,\varepsilon }- \zeta I)^{-1}$, $(\mathcal {A}_{N,\varepsilon }-\zeta I)^{-1}$ are obtained with error estimates depending on $\varepsilon$ and $\zeta$.