Homogenization for non-self-adjoint locally periodic elliptic operators

TL;DR

This paper develops homogenization approximations for non-self-adjoint elliptic operators with periodic coefficients, providing sharp error bounds for resolvent and gradient approximations in the context of locally periodic media.

Contribution

It introduces a homogenization framework for non-self-adjoint operators with periodic coefficients, including explicit two-term resolvent approximations and error estimates.

Findings

Two-term uniform approximation for resolvent operators

First-term approximation for the gradient of resolvent operators

Sharp-order bounds on approximation errors

Abstract

We study the homogenization problem for matrix strongly elliptic operators on $L_2(\mathbb R^d)^n$ of the form $\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$. The function $A$ is Lipschitz in the first variable and periodic in the second. We do not require that $A^*=A$, so $\mathcal A^\varepsilon$ need not be self-adjoint. In this paper, we provide, for small $\varepsilon$, two terms in the uniform approximation for $(\mathcal A^\varepsilon-\mu)^{-1}$ and a first term in the uniform approximation for $\nabla(\mathcal A^\varepsilon-\mu)^{-1}$. Primary attention is paid to proving sharp-order bounds on the errors of the approximations.