Homogenization of periodic differential operators of high order

Abstract:

A periodic differential operator of the form $A_\varepsilon = (\mathbf {D}^p)^\ast g(\mathbf {x} / \varepsilon ) \mathbf {D}^p$ is considered on $L_2(\mathbb {R}^d)$; here $g(x)$ is a positive definite symmetric tensor of order $2 p$ periodic with respect to a lattice $\Gamma$. The behavior of the resolvent of the operator $A_\varepsilon$ as $\varepsilon \to 0$ is studied. It is shown that the resolvent $(A_\varepsilon + I)^{-1}$ converges in the operator norm to the resolvent of the effective operator $A^0$ with constant coefficients. For the norm of the difference of resolvents, an estimate of order $\varepsilon$ is obtained.