Homogenization of elliptic systems with periodic coefficients: operator error estimates in $L_2(\mathbb {R}^d)$ with corrector taken into account

Abstract:

A matrix elliptic selfadjoint second order differential operator (DO) ${\mathcal B}_{\varepsilon }$ with rapidly oscillating coefficients is considered in $L_2(\mathbb {R}^d;\mathbb {C}^n)$. The principal part $b(\mathbf {D})^* g(\varepsilon ^{-1}\mathbf {x})b(\mathbf {D})$ of this operator is given in a factorized form, where $g$ is a periodic, bounded, and positive definite matrix-valued function and $b(\mathbf {D})$ is a matrix first order DO whose symbol is a matrix of maximal rank. The operator ${\mathcal B}_\varepsilon$ also includes first and zero order terms with unbounded coefficients. The problem of homogenization in the small period limit is studied. For the generalized resolvent of $\mathcal {B}_\varepsilon$, approximation in the $L_2(\mathbb {R}^d;\mathbb {C}^n)$-operator norm with an error $O(\varepsilon ^2)$ is obtained. The principal term of this approximation is given by the generalized resolvent of the effective operator ${\mathcal B}^0$ with constant coefficients. The first order corrector is taken into account. The error estimate obtained is order sharp; the constants in estimates are controlled in terms of the problem data. General results are applied to homogenization problems for the Schrödinger operator and the two-dimensional Pauli operator with singular rapidly oscillating potentials.