Homogenization in the Sobolev class $H^1(\mathbb R^d)$ for second order periodic elliptic operators with the inclusion of first order terms

Abstract:

Matrix periodic elliptic second order differential operators ${\mathcal B}_{\varepsilon }$ in $\mathbb {R}^d$ with rapidly oscillating coefficients (depending on $\mathbf {x}/\varepsilon$) are studied. The principal part of the operator is given in a factorized form $b(\mathbf {D})^* g(\varepsilon ^{-1}\mathbf {x})b(\mathbf {D})$, where $g$ is a periodic, bounded and positive definite matrix-valued function and $b(\mathbf {D})$ is a matrix first order operator whose symbol is a matrix of maximal rank. The operator also has zero and first order terms with unbounded coefficients. The problem of homogenization in the small period limit is considered. Approximation for the generalized resolvent of the operator ${\mathcal B}_\varepsilon$ is obtained in the operator norm in $L_2(\mathbb {R}^d;\mathbb {C}^n)$ with error term $O(\varepsilon )$. Also, approximation for this resolvent is obtained in the norm of operators acting from $L_2(\mathbb {R}^d;\mathbb {C}^n)$ to $H^1(\mathbb {R}^d;\mathbb {C}^n)$ with error term of order $\varepsilon$ and with the corrector taken into account. The general results are applied to homogenization problems for the Schrödinger operator and the two-dimensional Pauli operator with potentials involving singular terms.