Spectral approach to homogenization of an elliptic operator periodic in some directions

TL;DR

This paper develops a spectral method to analyze the homogenization of elliptic operators with periodic coefficients in some directions, providing convergence results and estimates for the inverse operators as the period tends to zero.

Contribution

It introduces a spectral approach to homogenization for elliptic operators with periodic coefficients, deriving operator norm convergence and sharp estimates for the inverse operators.

Findings

Operator $(A_ ext{ε}+Q^ ext{ε})^{-1}$ converges to $(A^0+Q^0)^{-1}$ as ε→0.

Established a sharp order estimate for the difference of inverse operators.

Applied results to homogenization of Schrödinger operators with periodic singular potentials.

Abstract

The operator \[ A_{\varepsilon}= D_{1} g_{1}(x_{1}/\varepsilon, x_{2}) D_{1} + D_{2} g_{2}(x_{1}/\varepsilon, x_{2}) D_{2} \] is considered in $L_{2}({\mathbb{R}}^{2})$, where $g_{j}(x_{1},x_{2})$, $j=1,2,$ are periodic in $x_{1}$ with period 1, bounded and positive definite. Let function $Q(x_{1},x_{2})$ be bounded, positive definite and periodic in $x_{1}$ with period 1. Let $Q^{\varepsilon}(x_{1},x_{2})= Q(x_{1}/\varepsilon, x_{2})$. The behavior of the operator $(A_{\varepsilon}+ Q^{\varepsilon}%)^{-1}$ as $\varepsilon\to0$ is studied. It is proved that the operator $(A_{\varepsilon}+ Q^{\varepsilon})^{-1}$ tends to $(A^{0} + Q^{0})^{-1}$ in the operator norm in $L_{2}(\mathbb{R}^{2})$. Here $A^{0}$ is the effective operator whose coefficients depend only on $x_{2}$, $Q^{0}$ is the mean value of $Q$ in $x_{1}$. A sharp order estimate for the norm of the difference $(A_{\varepsilon}+ Q^{\varepsilon})^{-1}- (A^{0} + Q^{0})^{-1}$ is obtained. The result is applied to homogenization of the Schr\"odinger operator with a singular potential periodic in one direction.