Operator estimates in homogenization of L\'evy-type operators with periodic coefficients

TL;DR

This paper studies the homogenization of Lévy-type operators with periodic coefficients, proving convergence of resolvents and providing explicit rates depending on the parameter alpha.

Contribution

It establishes the operator norm convergence of homogenized Lévy-type operators with periodic coefficients and derives explicit convergence rates based on alpha.

Findings

Resolvent convergence in operator norm as epsilon approaches zero

Explicit error estimates depending on alpha

Homogenized operator characterized by mean value of coefficients

Abstract

The paper deals with homogenization of self-adjoint operators in $L_2(\mathbb R^d)$ of the form $$ ({\mathbb A}_\eps u) (\x) = \int_{\R^d} \mu(\x/\eps, \y/\eps) \frac{\left( u(\x) - u(\y) \right)}{|\x - \y|^{d+\alpha}}\,d\y, $$ where $0< \alpha < 2$, and $\eps>0$ is a small parameter. It is assumed that the function $\mu(\x,\y)$ is $\Z^d$-periodic in each variable, $\mu(\x,\y)=\mu(\y,\x)$ for all $\x$ and $\y$, and $0< \mu_- \leqslant \mu(\x,\y) \leqslant \mu_+< \infty$. Under these assumptions we show that the resolvent $({\mathbb A}_\eps + I)^{-1}$ converges, as $\eps\to0$, in the operator norm in $L_2(\R^d)$ to the resolvent $({\mathbb A}^0 + I)^{-1}$ of the limit operator ${\mathbb A}^0$ given by $$ ({\mathbb A}^0 u) (\x) = \int_{\R^d} \mu^0 \frac{\left( u(\x) - u(\y) \right)}{|\x - \y|^{d+\alpha}}\,d\y, $$ where $\mu^0$ is the mean value of $\mu(\x,\y)$. We also show that the operator norm of the discrepancy $\|({\mathbb A}_\eps + I)^{-1} - (\A^0 + I)^{-1}\|_{L_2(\mathbb R^d)\to L_2(\mathbb R^d)}$ can be estimated by $O(\eps^\alpha)$, if $0< \alpha < 1$, by $O(\eps (1 + | \operatorname{ln} \eps|)^2)$, if $ \alpha =1$, and by $O(\eps^{2- \alpha})$, if $1< \alpha < 2$.