On operator estimates in homogenization of non-local operators of convolution type

Abstract

The paper studies a bounded symmetric operator ${\mathbf{A}}_{\epsilon }$ in $L_2({\mathbf{R}}^d)$ with $$({\mathbf{A}}_{\epsilon }u)(x)={\epsilon }^{-d-2}{\int }_{{\mathbf{R}}^d}a((x-y)/\epsilon )\mu (x/\epsilon ,y/\epsilon )\left(u(x)-u(y)\right)dy;$$ here $\epsilon$ is a small positive parameter. It is assumed that $a(x)$ is a non-negative $L_1({\mathbf{R}}^d)$ function such that $a(-x)=a(x)$ and the moments $M_k={\int }_{{\mathbf{R}}^d}|x{|}^{k}a(x)dx$, $k=1,2,3$, are finite. It is also assumed that $\mu (x,y)$ is ${\mathbf{Z}}^d$-periodic both in $x$ and $y$ function such that $\mu (x,y)=\mu (y,x)$ and $0<{\mu }_{-}\le \mu (x,y)\le {\mu }_{+}<\infty$. Our goal is to study the limit behaviour of the resolvent $({\mathbf{A}}_{\epsilon }+I{)}^{-1}$, as $\epsilon \to 0$. We show that, as $\epsilon \to 0$, the operator $({\mathbf{A}}_{\epsilon }+I{)}^{-1}$ converges in the operator norm in $L_2({\mathbf{R}}^d)$ to the resolvent $({\mathbf{A}}^0+I{)}^{-1}$ of the effective operator ${\mathbf{A}}^0$ being a second order elliptic differential operator with constant coefficients of the form ${\mathbf{A}}^0=-\mathrm{div}{g}^0\nabla$. We then obtain sharp in order estimates of the rate of convergence.