Quantitative Estimates on Reiterated Homogenization of Linear Elliptic operators Using Fourier Transform Methods

TL;DR

This paper develops Fourier transform methods to analyze reiterated homogenization of linear elliptic operators, providing error estimates and scale separation techniques for complex multiscale problems.

Contribution

Introduces Fourier transform techniques to handle multiple scales in reiterated homogenization, achieving $O( ext{epsilon})$ error estimates and extending to Neumann problems.

Findings

Error estimates of $O( ext{epsilon})$ for homogenization

Fourier transform effectively separates multiple scales

Method adaptable to higher-order reiterated homogenization

Abstract

In this paper, we are interested in the reiterated homogenization of linear elliptic equations of the form $-\frac{\partial}{\partial x_{i}} \left(a_{i j} \left(\frac{x}{\varepsilon}, \frac{x}{\varepsilon^{2}}\right) \frac{\partial u_\varepsilon}{\partial x_{j}}\right)=f$ in $\Omega$ with Dirichlet boundary conditions. We obtain error estimates $O(\varepsilon)$ for a bounded $C^{1,1}$ domain for this equation as well as the interior Lipschitz estimates at (very) large scale. Compared to the general homogenization problems, the difficulty in the reiterated homogenization is that we need to handle different scales of $x$. To overcome this difficulty, we firstly introduce the Fourier transform in the homogenization theory to separate these different scales. We also note that this method may be adapted to the following reiterated homogenization problem: $-\frac{\partial}{\partial x_{i}} \left(a_{i j} \left(\frac{x}{\varepsilon},\cdots, \frac{x}{\varepsilon^{N}}\right) \frac{\partial u_\varepsilon}{\partial x_{j}} \right) = f$ in $\Omega$ with Dirichlet boundary conditions. Moreover, our results may be extended to the related Neumann boundary problems without any real difficulty.