• DOI:10.1137/15M1049981
• Corpus ID: 45921344

Homogenization for Non-self-adjoint Periodic Elliptic Operators on an Infinite Cylinder

@article{Senik2017HomogenizationFN,
  title={Homogenization for Non-self-adjoint Periodic Elliptic Operators on an Infinite Cylinder},
  author={Nikita N. Senik},
  journal={SIAM J. Math. Anal.},
  year={2017},
  volume={49},
  pages={874-898}
}

• N. N. Senik
• Published 20 August 2015
• Mathematics, Computer Science
• SIAM J. Math. Anal.

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.