Homogenization of non-symmetric convolution type operators

TL;DR

This paper investigates the homogenization of non-symmetric convolution operators, providing sharp operator norm approximations of their resolvents and identifying the effective operator and correction terms.

Contribution

It introduces a method to approximate the resolvent of non-symmetric convolution operators with explicit error bounds, extending homogenization theory beyond symmetric cases.

Findings

Achieved $O( ext{eps})$ approximation of the resolvent in operator norm.

Derived the form of the effective operator ${ m div} g^0 abla$ and correction terms.

Established conditions under which the approximation holds with finite moments.

Abstract

The paper studies homogenization problem for a bounded in $L_2(\mathbb R^d)$ convolution type operator ${\mathbb A}_\eps$, $\eps >0$, of the form $$ ({\mathbb A}_\eps u) (\x) = \eps^{-d-2} \int_{\R^d} a((\x-\y)/\eps) \mu(\x/\eps, \y/\eps) \left( u(\x) - u(\y) \right)\,d\y. $$ It is assumed that $a(\x)$ is a non-negative function from $L_1(\R^d)$, and $\mu(\x,\y)$ is a periodic in $\x$ and $\y$ function such that $0< \mu_- \leqslant \mu(\x,\y) \leqslant \mu_+< \infty$. No symmetry assumption on $a(\cdot)$ and $\mu(\cdot)$ is imposed, so the operator ${\mathbb A}_\eps$ need not be self-adjoint. Under the assumption that the moments $M_k = \int_{\R^d} |\x|^k a(\x)\,d\x$, $k=1,2,3$, are finite we obtain, for small $\eps>0$, sharp in order approximation of the resolvent $({\mathbb A}_\eps + I)^{-1}$ in the operator norm in $L_2(\mathbb R^d)$, the discrepancy being of order $O(\eps)$. The approximation is given by an operator of the form $({\mathbb A}^0 + \eps^{-1} \langle \boldsymbol{\alpha},\nabla \rangle + I)^{-1}$ multiplied on the right by a periodic function $q_0(\x/\eps)$; here ${\mathbb A}^0 = - \operatorname{div}g^0 \nabla$ is the effective operator, and $\boldsymbol{\alpha}$ is a constant vector.