Homogenization of the Lévy-type Operators
Elena Zhizhina, Andrey Lvovich Piatnitski, Vladimir Anatolevich Sloushch, Tatiana Aleksandrovna Suslina · Functional Analysis and Its Applications · 2025
In $$L_2(\mathbb R^d)$$ , we consider a selfadjoint operator $${\mathbb A}_\varepsilon$$ , $$\varepsilon >0$$ , of the form $$({\mathbb A}_\varepsilon u) (\mathbf{x}) =\int_{\mathbb R^d} \mu\biggl(\frac{\mathbf{x}}{\varepsilon},\frac{\mathbf{y}}{\varepsilon}\biggr) \frac{(u(\mathbf{x}) -u(\mathbf{y}))}{| \mathbf{x}-\mathbf{y} |^{d+\alpha}}\,d \mathbf{y},$$ where $$0< \alpha < 2$$ . It is assumed that a function $$\mu(\mathbf{x},\mathbf{y})$$ is bounded, positive definite, periodic in each variable, and is such that $$\mu(\mathbf{x},\mathbf{y})=\mu(\mathbf{y},\mathbf{x})$$ . A rigorous definition of the operator $${\mathbb A}_\varepsilon$$ is given in terms of the corresponding quadratic form. It is proved that the resolvent $$({\mathbb A}_\varepsilon+I)^{-1}$$ converges in the operator norm on $$L_2(\mathbb R^d)$$ to the operator $$({\mathbb A}^0+I)^{-1}$$ as $$\varepsilon\to 0$$ . Here, $${\mathbb A}^0$$ is an effective operator of the same form with the constant coefficient $$\mu^0$$ equal to the mean value of $$\mu(\mathbf{x},\mathbf{y})$$ . We obtain an error estimate of order $$O(\varepsilon^\alpha)$$ for $$0< \alpha < 1$$ , $$O(\varepsilon (1+| \operatorname{ln} \varepsilon|)^2)$$ for $$ \alpha=1$$ , and $$O(\varepsilon^{2- \alpha})$$ for $$1< \alpha < 2$$ . In the case where $$1< \alpha < 2$$ , the result is refined by taking the correctors into account.