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 $\nabla_{x_{2}}(\mathcal{A}^{\varepsilon}-\mu)^{-1}$ for an appropriate $\mu$ converge in the operator norm to, respectively, $(\mathcal{A}^{0}-\mu)^{-1}$ and $\nabla_{x_{2}}(\mathcal{A}^{0}-\mu)^{-1}$, where $\mathcal{A}^{0}$ is an operator whose coefficients depend only on $x_{2}$. We also obtain an approximation for $\nabla_{x_{1}}(\mathcal{A}^{\varepsilon}-\mu)^{-1}$ and find the next term in the approximation for $(\mathcal{A}^{\varepsilon}-\mu)^{-1}$. Estimates for the rates of convergence ...