Homogenization of nonlocal convolution type operators: Approximation for the resolvent with corrector

Abstract

In $L_2({\mathbb{R}}^d)$, we consider a selfadjoint bounded operator ${\mathbb{A}}_{\epsilon }$, $\epsilon >0$, of the form $$({\mathbb{A}}_{\epsilon }u)(\mathbf{x})={\epsilon }^{-d-2}{\int }_{{\mathbb{R}}^d}a((\mathbf{x}-\mathbf{y})/\epsilon )\mu (\mathbf{x}/\epsilon ,\mathbf{y}/\epsilon )\left(u(\mathbf{x})-u(\mathbf{y})\right)d\mathbf{y}.$$ It is assumed that $a(\mathbf{x})$ is a nonnegative function of class $L_1({\mathbb{R}}^d)$ such that \hbox{$a(-\mathbf{x})=a(\mathbf{x})$} and $\mu (\mathbf{x},\mathbf{y})$ is ${\mathbb{Z}}^d$-periodic in each variable and such that $\mu (\mathbf{x},\mathbf{y})=\mu (\mathbf{y},\mathbf{x})$ and $0<{\mu }_{-}⩽\mu (\mathbf{x},\mathbf{y})⩽{\mu }_{+}<\infty$. Moreover, it is assumed that the moments $M_k(a)={\int }_{{\mathbb{R}}^d}|\mathbf{x}{|}^{k}a(\mathbf{x})d\mathbf{x}$, $k=1,2,3,4,$ are finite. We obtain approximation of the resolvent $({\mathbb{A}}_{\epsilon }+I{)}^{-1}$ for small $\epsilon$ in the operator norm on $L_2({\mathbb{R}}^d)$ with error of order $O({\epsilon }^2)$.

33 pages