Homogenization for non-self-adjoint periodic elliptic operators on an infinite cylinder

Abstract

We consider the problem of homogenization for non-self-adjoint second-order elliptic differential operators~${\mathcal{A}}^{\epsilon }$ of divergence form on $L_2({\mathbb{R}}^{d_1}\times {\mathbb{T}}^{d_2})$, where $d_1$ is positive and~$d_2$ is non-negative. The~coefficients of the operator~${\mathcal{A}}^{\epsilon }$ are periodic in the first variable with period~$\epsilon$ and smooth in a certain sense in the second. We show that, as $\epsilon$ gets small, $({\mathcal{A}}^{\epsilon }-\mu {)}^{-1}$ and~$D_{x_2}({\mathcal{A}}^{\epsilon }-\mu {)}^{-1}$ converge in the operator norm to, respectively, $({\mathcal{A}}^0-\mu {)}^{-1}$ and~$D_{x_2}({\mathcal{A}}^0-\mu {)}^{-1}$, where ${\mathcal{A}}^0$ is an operator whose coefficients depend only on~$x_2$. We also obtain an approximation for $D_{x_1}({\mathcal{A}}^{\epsilon }-\mu {)}^{-1}$ and find the next term in the approximation for~$({\mathcal{A}}^{\epsilon }-\mu {)}^{-1}$. Estimates for the rates of convergence and the rates of approximation are provided and are sharp with respect to the order.

Improved exposition, added new result (Theorem 3)