Two-parametric error estimates in homogenization of second order elliptic systems in ${\mathbb{R}}^d$ including lower order terms

Abstract

In $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$, we consider a selfadjoint operator ${\mathcal{B}}_{\epsilon }$, $0<\epsilon ⩽1$, given by the differential expression $b(\mathbf{D}{)}^{\ast }g(\mathbf{x}/\epsilon )b(\mathbf{D})+{\sum }_{j=1}^d(a_j(\mathbf{x}/\epsilon )D_j+D_j{a}_j(\mathbf{x}/\epsilon {)}^{\ast })+Q(\mathbf{x}/\epsilon )$, where $b(\mathbf{D})={\sum }_{l=1}^d{b}_l{D}_l$ is the first order differential operator, and $g,a_j,Q$ are matrix-valued functions in ${\mathbb{R}}^d$ periodic with respect to some lattice $\Gamma$. It is assumed that $g$ is bounded and positive definite, while $a_j$ and $Q$ are, in general, unbounded. We study the generalized resolvent $({\mathcal{B}}_{\epsilon }-\zeta Q_0(\mathbf{x}/\epsilon ){)}^{-1}$, where $Q_0$ is a $\Gamma$-periodic, bounded and positive definite matrix-valued function, and $\zeta$ is a complex-valued parameter. Approximations for the generalized resolvent in the $(L_2\to L_2)$- and $(L_2\to H^1)$-norms with two-parametric error estimates (with respect to the parameters $\epsilon$ and $\zeta$) are obtained.

59 pages

References in corpus (1)

• Interior error estimate for periodic homogenization