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~$\mathcal{A}^{\varepsilon}$ 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~$\mathcal{A}^{\varepsilon}$ are periodic in the first variable with period~$\varepsilon$ and smooth in a certain sense in the second. We show that, as $\varepsilon$ gets small, $(\mathcal{A}^{\varepsilon}-\mu)^{-1}$ and~$D_{x_{2}}(\mathcal{A}^{\varepsilon}-\mu)^{-1}$ 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 $D_{x_{1}}(\mathcal{A}^{\varepsilon}-\mu)^{-1}$ and find the next term in the approximation for~$(\mathcal{A}^{\varepsilon}-\mu)^{-1}$. Estimates for the rates of convergence and the rates of approximation are provided and are sharp with respect to the order.