On the Homogenization of Nonlocal Convolution Type Operators

Abstract

In \(L_2(\mathbb{R}^d)\), we consider a self-adjoint bounded operator \({\mathbb A}_\varepsilon\), \(\varepsilon >0\), of the form

It is assumed that \(a(\mathbf{x})\) is a nonnegative function such that \(a(-\mathbf{x}) = a(\mathbf{x})\) and \(\int_{\mathbb{R}^d} (1+| \mathbf{x} |^4) a(\mathbf{x})\,d\mathbf{x}<\infty\); \(\mu(\mathbf{x},\mathbf{y})\) is \(\mathbb{Z}^d\)-periodic in each variable, \(\mu(\mathbf{x},\mathbf{y}) = \mu(\mathbf{y},\mathbf{x})\) and \(0< \mu_- \leqslant \mu(\mathbf{x},\mathbf{y}) \leqslant \mu_+< \infty\). For small \(\varepsilon\), we obtain an approximation of the resolvent \(({\mathbb A}_\varepsilon + I)^{-1}\) in the operator norm on \(L_2(\mathbb{R}^d)\) with an error of order \(O(\varepsilon^2)\).

DOI 10.1134/S106192084010114