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

TL;DR

This paper studies homogenization of non-self-adjoint elliptic operators on an infinite cylinder, showing operator norm convergence of resolvents and derivatives as the periodicity parameter tends to zero, with sharp convergence rates.

Contribution

It provides the first detailed analysis of homogenization for non-self-adjoint operators on infinite cylinders, including explicit convergence rates and higher-order approximations.

Findings

Operator resolvents converge in norm as periodicity vanishes.

Derived explicit rates of convergence and approximation.

Established higher-order correction terms for the homogenized operator.

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.