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

$$({\mathbb A}_\varepsilon u) (\mathbf{x}) = \varepsilon^{-d-2} \int_{\mathbb{R}^d} a((\mathbf{x} - \mathbf{y} )/ \varepsilon ) \mu(\mathbf{x} /\varepsilon, \mathbf{y} /\varepsilon) \left( u(\mathbf{x}) - u(\mathbf{y}) \right)\, d\mathbf{y}.$$

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