On the Homogenization of Nonlocal Convolution Type Operators
Andrey Lvovich Piatnitski, Vladimir Anatolevich Sloushch, Tatiana Aleksandrovna Suslina, E. Zhizhina · Russian Journal of Mathematical Physics · 2024
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