On Homogenization for Non-Self-Adjoint Periodic Elliptic Operators on an Infinite Cylinder

We consider an operator $\mathcal{A}^{\varepsilon}$ on $L_{2}(\mathbb{R}^{d_{1}}\times\mathbb{T}^{d_{2}})$ ($d_{1}$ is positive, while $d_{2}$ can be zero) given by $\mathcal{A}^{\varepsilon}=-\operatorname{div} A(\varepsilon^{-1}x_{1},x_{2})\nabla$, where $A$ is periodic in the first variable and smooth in a sense in the second. We present approximations for $(\mathcal{A}^{\varepsilon}-\mu)^{-1}$ and $\nabla(\mathcal{A}^{\varepsilon}-\mu)^{-1}$ (with appropriate $\mu$) in the operator norm when $\varepsilon$ is small. We also provide estimates for the rates of approximation that are sharp with respect to the order.